Claiming for second French citizen.

Hello.
My second baby was born on 4th February 2018, mother of bay is French national. I am pakistani national with a 5 year resident permit valid march 2020.

We are claiming for child benefit, for first baby, every thing is coming OK, how we register for second baby, mum and baby in France, at moment. Mum is on maternity leave she work in UK..

Plz share your views..
Also, will this only benefit affect me in way when I apply for my Permanent residence in 2020, plz share your views.. This benefit business is all under my wife name...