ought to help with

He ought to help with the housework once a week.

help (intransitive verb) = 幫助, It takes no object.

with (preposition) ~ used when considering one fact in relation to another
with (前置詞) 的功能 ~ 考慮到一事 (help) 與某事 (housework) 的關係
help是關係什麼事 (relating to what ), 要使用前置詞 with 敘述有關 housework

Help with something (幫助做某事)
~ to make it possible or easier to do something, by doing part of the work yourself, or by giving advice, money or support etc.

Compare:
Help someone with something
Help ~ transitive verb
Someone ~ object