Quote:They have moved into a flat that we helped them to paint and decorate.
Since the sentence is about ". . . a flat" that the speaker's group ". . . helped to paint and decorate," then the word 'them' is unnecessary.
As for the preposition 'to,' it may be omitted from the sentence, but it would be best to retain it as it would help to clarify any ambiguity that may arise in the sentence if it were to be left out.
They have moved into a flat that we helped to paint and decorate.
They have moved into a flat that we helped paint and decorate.