얘길 하고

talk about

→ 얘기+를
phrase Common #2,578
talk about 얘길 is a contracted form of 얘기를 (“話” + object marker). 하고 often connects the phrase to what follows, as in 얘길 하고 싶어요 (“want to talk about”).
KO: 오늘은 여행 얘길 하고 싶어요.
EN: I want to talk about travel today.

지금까지 쭉 그 얘길 하고 있었잖아요?

We've been talking about that all along, right?

— 희생 (1986)

얘길 하고 싶어해요

They want to talk.

— 좋은 친구들 (1990)

사고 얘길 하고 밤새 파티도 했다고 했다

They said they talked about the accident and partied all night.

— 좋은 친구들 (1990)