Cleaning oneself with toilet paper to remove madhiy; is it necessary to clean oneself (istinja’) with water first in order for wudu to be valid?

Madhiy is najis (impure) according to scholarly consensus. 

An-Nawawi (may Allah have mercy on him) said: The ummah is unanimously agreed that madhiy (prostatic fluid) and wadiy (a thick white fluid which is passed after passing urine) are naajis.

End quote from al-Majmoo‘ Sharh al-Muhadhdhab, 1/ 571 

Secondly: 

Tissues, cloth and stones are sufficient to remove impurities that are on the body, whether they are in the front or back passage or anywhere else on the body. 

There is no differentiation between regular impurities such as urine and stools, and irregular impurities such as wadiy and madhiy. 

But it is better to clean oneself (istinja’) with water from unusual emissions, acting in accordance with the Sunnah and so as to avoid an area of scholarly differences of opinion, and because that is more effective in cleansing and purifying. 

Al-Bahooti (may Allah have mercy on him) said: it is obligatory to cleanse oneself with water (istinja’) or with stones and the like (istijmaar) from everything that regularly comes out of the front and back passages, such as urine or stools, because Allah, may He be exalted, says (interpretation of the meaning): “And keep away from all impurity” [al-Muddaththir 74:5]; this includes every part of the clothing and the body. And the Prophet (blessings and peace of Allah be upon him) said: “If one of you goes to relieve himself, let him take three stones with him, and they will be sufficient for him.” Narrated by Abu Dawood; the command here indicates that it is obligatory. He said that it is sufficient, which clearly indicates that it is obligatory.

End quote from Kashshaaf al-Qinaa‘ ‘an Matn al-Iqnaa‘, 1/70 

It says in al-Mawsoo‘ah al-Fiqhiyyah (4/117): Madhiy is najis (impure) according to the Hanafis and it is something for which one must clean oneself (istinja’) with water or with stones. 

It is acceptable to clean oneself of it using stones or water; this is also a view among the Maalikis that is different to their well-known view. This is also the more correct view according to the Shaafa‘is, and is mentioned in one report from the Hanbalis. 

The well-known view of the Maalikis, which is also the other view narrated from the Hanbalis, specifies that water must be used and using stones etc is not valid, because of the report which says that ‘Ali (may Allah be pleased with him) said: “I was a man who used to emit a great deal of madhiy, and I felt too shy to ask the Messenger of Allah (blessings and peace of Allah be upon him) about that because I was married to his daughter, so I told al-Miqdaad ibn al-Aswad to ask him. So he asked him and he said: “Let him wash his penis and testicles and do wudu.” According to another version, he said: “Let him wash his penis and do wudu.” Ghusl is required for this, according to the Maalikis, only if it was emitted with pleasure; if it was not emitted with pleasure, then it is sufficient to use stones etc., so long as it does not occur every day like incontinence, in which case there is no requirement to remove it with water or stones; rather he is pardoned. End quote. 

But in order to cleanse oneself of madhiy it is essential to wash the testicles even if no madhiy got on them. See the answer to question no. 2458 

If there is any najaasah on one’s clothing, it is essential to wash it, and it is not sufficient to wipe it with a tissue. 

To sum up, madhiy may be purified using tissue or otherwise, but it is essential to wash the testicles according to the more correct opinion, and it is essential to wash off with water whatever has got onto the clothes. 

If you did wudu and forgot to cleanse yourself of madhiy (and you did not clean yourself with water or stones etc, and you did not wash it off your clothes), you do not have to repeat wudu; rather you should cleanse yourself of the impurity only, and pray, because cleansing yourself and removing the impurity before doing wudu is mustahabb (recommended), but it is not obligatory according to the more correct view. 

See al-Mughni by Ibn Qudaamah (1/78); al-Mawsoo‘ah al-Fiqhiyyah (4/115) 

Touching the private part without desire does not invalidate wudu. 

And Allah knows best.