won't stop giving cash in hand

I need advice... so my partner has a 4 year old with Someone else. she is a piece of work and most of the time I don't get involved.. however we've been working with people in the system to help us get on track with our legal rights towards all of this. I'm my country the father has no legal right unless the mother signs a sheet of paper saying she gives him legal rights. anyways we pay her money every week... she refuses to give us her bank details or any possible way of having a paper trail.. if he asks her to set one up she says well see.. and then demands the money cash in hand. my partner infuriates me because he just does whatever she asks and never thinks about the future or our son and how this would affect us if it ever went to court. we've been paying for the last 4 years and we have no paper trail... we've been working with a lady in the system (voluntary I asked for help because of all this mess) she said to my partner to refuse to pay BM until she gives an account so we can have a paper trail. we have no court arrangement so the money he gives is an arrangement between them. he agreed and then when this lady came to check up on us after 2 weeks he says to us he has told her he's not paying her till she gives him an account to pay... then I found out last week that he's just lied to our faces and been paying her the whole time. we don't want to go to court because to be honest the arrangement set up is suiting us all... it's just we need a paper trail and he keeps paying her because she says he can't see his daughter and were back to square one..