Having corrected the grid, thanks to Lie-In King, I now find there are several clues I can't solve.
Babylonian demi-god or first man: ?d?p?
Martial arts weapon: ???c?a??
In Greek mythology, a nymph of glens and groves: a?s?i?
Anglo Saxon poet and scholar: ?y?e???f
Island in the Lesser Antilles: A???a