Daughter and boyfriend don't get along...

Hello, I hope I can get some input from you guys here, I'd really appreciate it.

Last year I started a relationship with this guy from UK, (I'm mexican) and ever since he and my 6 yr old daughter have not gotten well at all. At first it was nice but when we (as a couple) started discipline things went downhill, I knew she needed this as all her life she had been spoiled by her grandma and dad. I'm a single mum working full time so every time I came home I had to deal with what they did the rest of the day (spending only a couple of hours a day together, I was not much of a help), but to be honest she's improved a lot since my bf came into our lives. The problem is there is still hostility in their relationship whenever they see each other, which is not that often, maybe a couple of times a week.

I believe the whole language barrier doesn't help either, he has learned lots of spanish and she does ok with english, but this makes fertile ground for many misunderstandings. There was even a time when she said that she chose not to do as asked because she didn't like him, and that she thought if she continued like this we would eventually break up. This has definitely damaged my relationship both with him and my kid.

I try to be the "not so bad cop" but I have caught myself under minding what he does and she thinks that if he tells her off, mummy will come to the rescue, I accept that's my fault. I have tried to talk some sense into him, but he says she needs to change and do as asked otherwise she "wins". He says that if we work together, she will realize she will have no other choice but to comply and behave. I have told him that we as adults have to make the change but he'll have none of it.

I'm afraid that if we someday get married things will only get worse. And I'm also afraid that if we do break up, this will become an even bigger problem once she hits the teen years...

Thanks in advance!