【give someone a foot up】 —help a person in time of need:

• If we can’t help them, we try to recommend them to another agency. Our motto is to give a foot up and not a handout.

Note: The expression does not correlate in meaning with the phrase give someone a foot in the door—let smb. start working at a low level in the hope of getting a better position in the future:

• The freelance work that I did gave me a foot in the door.

【give someone a leg up】 —

1. help smb. to mount a horse or to climb up smth.:

• The wall is very low, sir, and your servant will give you a leg up.

2. = 【give someone a foot up】 :

• Keep on your regular patients and give that young dentist a leg up by recommending any newcomers to him.

Note: The expression does not correlate in meaning with the phrase give someone leg—(sl.) deceive or fool a person:

• Last time I saw you, you’re giving me a little leg about there’s nothing going on.