Learn English – A single word describing a person who is desperate for love/approval, who “comes running” if someone gives the smallest hint of being interested

Subject
The person being depicted or portrayed by a painter, an artist, or in this case an impersonator.

The impersonator's favourite subjects were politicians and well-known singers.

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One of Britain's most famous and loved impersonators during the 1970s and early '80s was Mike Yarwood. Yarwood enjoyed stratospheric popularity in the years when British television only had three channels. In the wikipedia page dedicated to the entertainer the term, subjects, is used.

Most of his most famous subjects, such as Heath and Wilson, retired from public life or died and he was unable to master new prominent figures, most significantly, the country's first woman Prime Minister, Margaret Thatcher.

Since a hitman is sort of a self-employed individual, the person giving the assignment, or contract, could be considered the client.