The term 'Baby Sitter's HANDFUL' refers to the difficult job of taking care of a baby or young child. The crossword solutions for this term are IMP (3 letters) and BRAT (4 letters). An IMP is a mischievous child, often described as a 'little devil, ' while a BRAT is a child who is spoiled and behaves badly. Both terms are used to describe children who can be difficult to manage, making them a 'HANDFUL' for a baby sitter.