First Tradition:

Imam al-Haithami recorded this tradition (link):

رأيت رسول الله صلى الله عليه و سلم فرج ما بين فخذي الحسين و قبل زبيبته
رواه الطبراني و إسناده حسن

I saw the Messenger of Allah pbuh putting Husein’s legs apart and kissing his penis.
Related by Al-Tabarani and it's chain of narration is GOOD (حسن).

Imam al-Dhahabi recorded similar tradition in his book History of Islam, and declared it's narrator Qabus to be Good in Hadith (link):

وَقَالَ جَرِيرُ بْنُ عَبْدِ الْحَمِيدِ ، عَنْ قَابُوسَ ، عَنْ أَبِيهِ ، عَنِ ابْنِ عَبَّاسٍ: أَنّ رَسُولَ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ " فَرَّجَ بَيْنَ فَخِذَيِّ الْحَسَنِ وَقَبَّلَ زَبِيبَتَهُ " قَابُوسُ : حَسَنُ الْحَدِيثِ

Jarir from Qabus bin Abi Zayban, he from his father, who from Ibn Abbas: I saw the Messenger of Allah pbuh putting Hassan’s legs apart and kissing his penis. And Qabus bin Abi Zayban is Good in Hadith (حَسَنُ الْحَدِيثِ)

In Arabic, the word that is used for 'penis' is 'ذکر'. But in this tradition, the word 'زبيبته' has been used for penis, which is considered as a vulgar word for penis in Arabic language. 

Anyhow, this tradition became a problem for Islam apologists. And they tried to use the same tactic which they use every time in such situations i.e. they declared this tradition to be 'weak ضعيف'. They developed this so-called 'Ilm-ul-Hadith' in order to use it as a tool to discard all those Ahadith, which question the honor of Islam. 

Nevertheless, a lie always causes some 'contradictions'. This same contradiction happened with this Hadith too. Islam apologists claimed that one narrator Qabus bin Abi Zayban is declared weak. But they don't tell that many of Hadith Scholars declared this same person Qabus bin Abi Zayban to be Trustworthy too and accepted his Ahadith. Let us see those Scholars who accepted his Ahadith (link):

Qabus Ibn Abi Zabyan قابوس بن أبي ظبيان

قال محمد بن أحمد الذهبي: قابوس حسن الحديث

Muhammad ibn Ahmad Adh-Dhahabi says: “Qabus is good (Hasan) in traditions.” [Tarikh-ul-Islam, Vol. 4 page 36.]

قال نور الدين الهيثمي: وإسناده حسن

Nur-ud-Din Al-Haithami says: “Its chain of transmission is good (Hasan).” [Majma’-uz-Zawa’id, Hadith 15108.]

قال يحيى بن شرف النووي: صحيح

Yahya Ibn Sharaf An-Nawawi says: “Authentic (Sahih).” [Al-Majmu’, Vol. 2 page 47.]

قال إسلاموب نت: مقبول

Islamweb.net says: “Accepted (Maqbul).” [Al-Mu’jam-ul-Kabir, Number 2590.]

قال محمد بن عيسى الترمذي: حسن غريب

Muhammad Ibn Isa At-Tirmidhi says: “Good (Hasan), strange (Gharib).” [Sunan At-Tirmidhi, Hadith 1053.]

قال الحاكم النيسابوري: هذا حديث صحيح الإسناد ولم يخرجاه

Al-Hakim An-Naisaburi says: “The tradition is authentic (Sahih) according to its chain, but the two (Al-Bukhari and Muslim) did not publish it.” [Al-Mustadrak Ala As-Sahihain, Hadith 6995]

قال ابن حجر العسقلاني: أخرجه الطَبراني وفيه دليلٌ على أن الصَبي ليست له عروة

Ibn Hajar Al-Asqalani says: “At-Tabarani reported it and it is a proof, that a little child has no nakedness to cover.” [Ad-Dirayah Fi Takhrij Ahadith Al-Hidayah, page 124]

Second Tradition:

Any how, this same tradition was also recorded from another chain of narrators by Ibn Abi Layla. Imam Bayhiqi recorded it in his book (link):

أنبأ أبو بكر القاضي وأبو سعيد بن أبي عمرو قالا نا أبو العباس محمد بن يعقوب ثنا محمد بن إسحاق ثنا محمد بن عمران حدثني أبي حدثني بن أبي ليلى عن عيسى عن عبد الرحمن بن أبي ليلى قال : كنا عند النبي صلى الله عليه و سلم فجاء الحسن فأقبل يتمرغ عليه فرفع عن قميصه وقبل زبيبته

Abdul Rehman said: We were with the prophet (saw) when Hassan came to him. Then the messanger of Allah lift up his (al Hassan’s) shirt and kissed his penis. (al-Bayhiqi declared that it's chain of narrators is not strong)

All the narrators of this Hadith are trustworthy, except for Muhammad ibn Abi Layla. 

Nevertheless, none blamed Muhammad ibn Abi Layla to be a liar. At maximum, they objected that his memory is not strong. But according to their own rules of Ilm-ul-Hadith (i.e. Science of Hadith), a Hadith cannot be rejected completely only one the basis of weak memory of one narrator. A weak memory can cause small mistakes like change of names e.g. taking name of Hassan instead of Hussain etc, but the whole incident is not rejected. Especially in such cases, when the same tradition has also been recorded by ANOTHER chain of narrators too (for example, Qabus also narrated same tradition from another chain of narration). Both of these two Ahadith, with entirely different chain of narrators are supporting and thus strengthening each other and together they bring it to the level of Sahih (authentic) Hadith. 

That is why Ibn Kathir wrote in his book al-Badaya wa al-Nahaya (link):

وهو أكبر ولد أبويه وقد كان رسول الله (صلى الله عليه وآله) يحبه حباً شديداً حتى كان يقبل زبيبته وهو صغير وربما مص لسانه واعتنقه وداعبه.

Hassan was the eldest one among his siblings and the messenger of Allah loved him so strongly that he used to kiss his penis. And while Hassan was a little boy, that is why he also sucked his penis with his tongue. 

This Hadith is even part of Islamic Fiqh

Muslim Scholars of Fiqh use only those Ahadith in Fiqh Rulings, which they consider to be authentic enough for this purpose. And they indeed considered this Hadith of kissing the penis of Hassan to be strong enough to be used in Fiqh Rulings i.e. they allow in general to kiss penis of a small boy. 

Ibn Qadama wrote in his Fiqh book 'Al-Mughani' (link):

وعن الزهري و الأوزاعي لا وضوء على من مس ذكر الصغير لأنه يجوز مسه والنظر إليه وقد روي عن النبي صلى الله عليه و سلم أنه قبل زبيبة الحسن وروي وروي أن النبي صلى الله عليه و سلم مس زبيبة الحسن ولم يتوضأ

According to Imam Zuhri and Imam al-Owza'ii it is not needed to do Wadhu (ritual washing) again after touching the penis of a small boy, and it is also also allowed to see and touch it. And it is narrated from the Messanger of Islam (saw) that he kissed the penis of al-Hassan and he didn't do his Wadhu again. 

Ibn Taymiyyah wrote in his Fatwa book 'مجموع الفتاوی' (link):

قال أحمد بن تيمية: وقد كان النبي صلى الله عليه وسلم يقبل زبيبة الحسن

Ahmad bin Taymiyyah said: The Messanger of Allah (saw) used to kiss the penis of al-Hassan.

And Imam al-Nawawi wrote in his Fiqh Book (link):

قال يحيى بن شرف النووي: وقال بعض أهل العلم ينقض مسه ذكر نفسه دون غيره واحتج لهؤلاء بحديث طلق بن علي رضي الله عنه أن النبي صلى الله عليه وسلم سئل عن مس الذكر في الصلاة فقال: هل هو إلا بضعة منك وعن أبي ليلى قال كنا عند النبي صلى الله عليه وسلم فأقبل الحسن يتمرغ عليه فرفع عن قميصه وقبل زبيبته ولأنه مس عضوا منه فلم ينقض كسائر الأعضاء واحتج أصحابنا بحديث بسرة وهو صحيح كما قدمنا بيانه وبحديث أم حبيبة قالت: سمعت رسول الله صلى الله عليه وسلم يقول: من مس فرجه فليتوضأ قال البيهقي: قال الترمذي سألت أبا زرعة عن حديث أم حبيبة فاستحسنه قال ورأيته يعده محفوظا

“Some of the people of knowledge say: ,,The ablution is broken by touching one’s own penis excluding the other.” And I reply to these people with the proof of the tradition of Talq Ibn Ali, may God be pleased with him, that the Messenger (s.) was asked about touching from the penis in prayer, whereupon he said: “Should it be something other than a piece of you?” And through Abu Lailah, who said: “We were with the Prophet Muhammad (s.), whereupon Al-Hasan came rolling on him. Then he removed his shirt and kissed his penis.” And because he touched a part of his body,the ablution was not broken like the rest of the body parts and our companions prove it with the narration of Busrah and it is authentic (Sahih) just as we have submitted its explanation and in a narration Umm Habibah said: “I heard the Prophet Muhammad (s.) say: “Whoever touches his own private parts should perform the ablution.” Al-Baihaqi commented that At-Tirmidhi said: “I asked Abu Zur’ah about the narration of Umm Habibah, whereupon he considered it good (Hasan).” He said: “I saw him considering it as being preserved.” [Al-Majmu’, Vol. 2 pp. 46 – 47]

Thus:

– An-Nawawi takes the report as evidence
– He does not reject the intimate kiss
– He accepts that the Prophet (s.) kissed the part
– He explains the intimate kiss in accordance with ablution
– He proves the evidence through his companions
– He confirms that it is authentic (Sahih)(link)