Has left will be the correct option. Someone has left their handbag in the room.
What is the meaning of left in the sentence?
The term "left" means something or some part of it still remains when the rest has gone or been used.
What is handbag?
The term "handbag" means a small piece of luggage/bag that can be carried by hand.
It is so because the sentence is in present perfect form and with this form the suitable option will be "has left" as subject+ has/have+ participle is the formula for present perfect.
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