I read somewhere that Imam Mahdi atfs repents on our behalf twice a week. Is this authentic?
If yes which are those 2 days?

What does Repent meant? Yes he asks for forgiveness from Allah for us when he sees our Amal (deeds) are not correct. All these has mentioned in rewayat, also it has said that our Amal are presented to him on every Thursday and also on the night of Qadr. But when you say he repents ( means Tuaba) then there is a question that how come other than a person who has done wrong anyone else does Tauba.
We should know that this Duniya is a place where we have been sent to become a perfect Human being.
For becoming perfect human being we have to follow the some instructions just like if anyone wants to become a perfect surgeon he/she has to learn a lot and then most importantly has to work practical because just having theoretical knowledge can’t make him/her perfect in their profession.
We are here to do many things practically in the light of Divine guidance and leadership but WE HAVE TO DO not others even Prophet and Imam can’t do action (Amal) in behalf of us.
Same way when Prophets can’t do actions on behalf of us then of course they can’t REPENT on behalf of us it we ourselves we have to REPENT for our sin and for our mistakes.
Therefore it’s totally wrong and illogical concept that Imam e Zamana (ajtf) Repents on behalf of us. It doesn’t make sense if I have taken money from someone then why my brother has to pay for it no it we who has to pay. Yes Imam can ask Forgiveness for his followers as all Prophet (a.s) they asked Allah (swt) to forgive their nation (ummat) but again it’s up to Allah(swt) either He forgives or not it’s up to Him.
So unfortunately some people misguide people that don’t worry feel free and do whatever you want and Imam Zamana (ajtf) will repent for you. It’s nothing totally a concept of Christians as they say Jusus was Crucifiedto repent their Sins and so now they are relaxed and free to do whatever they want to do.