Here are the clues with the definition and the wordplay separated by a vertical line ( | ).
Words that hint at the type of wordplay are shown in italics.

Across:
1. That woman | is in ashes (3)
4. Legume | confuses large primate (3)
5. Painting | Mr. Carney [two definitions of the same word] (3)

Down:
1. Health club | starts to sell pet animals (3)
2. That woman | is nearly here (3)
3. Consume | Earl Grey start to finish (3)