手利き is read as てきき (tekiki) and means skill.
てきき
tekiki
てがきく
to be able
ききて
one's dominant hand
てつづき
procedure
てびき
guidance
てまねき
beckoning
てぬき
intentional negligence
てがき
handwriting
しゅずつ
skill with one's hands
てかず
number of moves (in go, shogi, etc.)
てあて
advance preparation
てがかり
handhold
しゅだん
means