From 上目(うわめ) (uwame) + 遣(つか)い (tsukai, “連(れん)用(よう)形(けい) (ren'yōkei, “stem or continuative form”) of the verb 遣(つか)う (tsukau).”). The tsukai changes to zukai as an instance of rendaku (連濁).
うわめづかい • (uwamezukai) ←うはめづかひ (ufamedukafi)?