This wasn't my perfect evening out, and I would like to make a claim for half the money back.

claim 是名词，意思是"索赔、赔偿要求"，指正式向机构或商家提出的、要求退还款项的书面申请。在这里 make a claim 表示"提出索赔"，即要求退还一半的钱。