From 使(つか)い (tsukai, “連(れん)用(よう)形(けい) (ren'yōkei, “stem or continuative form”) of the verb 使(つか)う (tsukau).”) + 勝(かっ)手(て) (katte). The katte changes to gatte as an instance of rendaku (連濁).
つかいがって • (tsukaigatte)