I read the following in a TOEIC book:

“1. In the future, the company will not let its part-time and temporary employees ______ overtime.

A. to work

B. be working

C. worked

D. work

(Answer: D)” ((Longman Preparation Series for the New TOEIC Test, Advanced Course, 4th Edition, p. 138))

I agree with the above that D is correct.

If there were ‘E. working’, would it be correct, also?

I don’t think ‘working’ would be correct, because I’ve never seen the structure “let + sb + doing sth” in any context.

What do you think? Do you say “let sb doing sth”

(Besides, do you say “let + sth + done”?)

Thanks in advance.
No, "working" would not be correct. More generally, "let sb doing sth" is incorrect.

"let sth done" is not possible, but "let sth be done" is (as in "He let it be known...").

You are right.

But we can say.....'Let us watch John painting the picture'.

Thank you.