After we had finished [having coffee], we went to the check-in.

having coffee 是动词 have 的现在分词形式，have coffee 意思是"喝咖啡"。 check-in 是由动词短语 check in（登记、办理手续）通过连字符连接形成的复合名词，指"办理登机手续的地方或过程"。