To be on a "sticky wicket" means to be in a difficult situation. It's a
figure of speech in the
English language, an
idiom. The dictionary lists it as a
noun, but it's also the object of a
prepositional phrase.
A wicket was originally (and can still be) a small door or grille, especially one cut into a larger door. The game of cricket borrowed the term to use for the three wooden sticks topped by two wooden bails. The bowler, similiar to a pitcher in baseball, throws the ball at the wicket and the batsman must defend the wicket from being hit. It's a base for the opposing team's batsmen. Croquet also has wickets, but they are wire arches.
In a usual double-ended game there are two sets of wickets, 22 yards apart, called a chain. By a further extension, the word came to apply to the ground between them (not entirely sure why, a grille within a grille?). After it rains, the ground on the pitch (the field) becomes soft and the ball bounces more erratically on the ground, making it more difficult for the batsman to hit. Hence a sticky wicket.
It is said to be very difficult to bat on a sticky wicket. So figuratively speaking, to be on a sticky wicket is to experience great difficulty.
Example: Cynthia is a sticky wicket. She won't help me write the paper.