It starts about a woman who is a drug lords squeeze and the drug lords hit man. As a payment to the hitman the drug lord gives the woman to the hitman for a day. The woman leaves with the drug lords money and the hit man is then hired to find her.

Is it a romance, maybe a paperback, like a Harlequin?

It was suspense-romance, but I'm not sure if it was Harlequin. Possibly.

I hate to make you work harder and wish I had an answer for you, but I'm going to have to suggest you call/visit your local public library and ask for the reader's advisory person or a reference librarian to help. The library has database and print resources to pin down a title and author based on bits of information you provide. If I remember correctly, there is even a paperback romance/suspense/cozy mystery reference source.

If that doesn't work for you, let me know and I will pin down my librarian friends. (I'm retired now, so don't have those sources here at home.)

Death Angel by Linda Howard