noun
A favor or request one asks of someone; often implies a somewhat burdensome or personal request.
ちょっと頼みごとがあるんだけど、今いい？
I have a small favor to ask — is now a good time?
彼は友達からの頼みごとを断れない。
He can't refuse a request from a friend.
お願い is a more general and polite word for a request or wish, while 頼みごと specifically implies a favor that may be somewhat demanding.
Compound of 頼み (request) + 事 (thing, matter).