A request or favor asked of someone. Often used in phrases like 頼みがある (I have a favor to ask) or 頼みを聞く (to grant a request).
ちょっと頼みがあるんだけど。
I have a small favor to ask.
He willingly accepted his friend's request.
The act of relying on or trusting someone or something. Often used in set phrases like 頼みとする (to rely on) or 頼みになる (to be dependable).
彼は私の唯一の頼みだ。
He is my only hope/reliance.
いざという時は君を頼みにするよ。
I'll count on you when the time comes.
I have a small favor to ask, is that okay?
He relies on his older brother to get by.
He granted his friend's request and lent him money.
This tool becomes reliable in times of disaster.
She refused her boss's request.
The teacher accepted the student's request.
頼み can mean 'reliance/support' just like 頼り, but it also commonly means 'a request', making 頼み slightly broader in usage.
頼み is a casual, everyday word for a favor or request between people, while 願い in this sense sounds more earnest or formal.
relying on strength; dependence on force
indirect request; request through an intermediary