faint hopes; dim prospects; unlikely to succeed
Describes a situation where the chance of success or a desired outcome is very low. Often used in the pattern 望み薄だ or 望み薄な.
The prospects for this plan's success are dim.
It's an unlikely situation that he will pass the exam.
Compound of 望み (hope) and 薄 (thin, weak). The exact historical derivation is uncertain, but the meaning is transparent from its components.