Just take the birth certificate with you
If the father has PR you have to have his permission. If he doesn't grant it you'll have to apply to court
@Nicole unless you have a "lives with" order in your favour, in which case you can take your child abroad for around a month without the other parents permission
I’ve done it 3 times and every single time I’ve only been asked for my son’s birth certificate coming back into the UK but in my experience it’s 10000% worth having the birth certificate. The man told me when coming back into the country last year that even just a picture on your phone is sufficient but I’ve always had a paper copy with me. With regards to a letter from dad, I’ve always taken one but never been asked for it. They seem to just ask for the birth certificate x