Programmatically remove items from cart

Is it possible to remove items from a users basket from code. My client has 2 free gift tiers one at ￡80 and one at ￡140. The gifts are different but they want to offer one gift at ￡80 and a different set at ￡140.

Ideally they want to remove the gift they got at ￡80 if the basket goes over the 2nd tier threshold.

Hope that makes sense.

Can items be removed programmatically from the basket?

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