My legal responsibilities as a landlord and council letter

FBaby

which essentially is the mother and her 3 little kids

What do you mean by 'essentially'? Either it's her with three young children or there is also another family staying.

Don't freak out. Write a letter and explain that you are in contact with the tenant regularly, that you have done inspection of X dates and at each time, the bedrooms were those of one adult and three kids, and have seen no evidence of more people living there.

Say that upon their letter, you've asked your tenant if you could come and take photographs, and that they've agreed because they are happy to fully co-operate as they know no-one else is living there.

They ask if they could guide you with any further evidence you could submit to show that the allegation that another family lives there is incorrect to the best of your knowledge.