Select the phrase/idiom that will improve the underlined part of the given sentence.
After two hours of wandering in the desert, we realised we were on a wild horse race.

The correct option to replace the underlined part is wild goose chase.

Explanation:
An idiom is a phrase or expression whose meaning cannot be understood from the literal meanings of its individual words.
In the given sentence, the phrase "wild horse race" is incorrect in the context of being lost or hopelessly searching for something in the desert.

The correct idiom is "wild goose chase", which means a foolish, hopeless, or unprofitable search for something that is unattainable or does not exist. Wandering aimlessly in a desert for two hours fits this meaning perfectly, as the search or direction turned out to be completely useless and futile.

Meaning of the idioms/phrases:
• Wild goose chase: A foolish and hopeless search or pursuit.
• The other options (wild jackal hunt, wild carpet rug, wild shadow chase) are not standard English idioms.

Therefore, replacing the underlined phrase gives the grammatically and idiomatically correct sentence:
"After two hours of wandering in the desert, we realised we were on a wild goose chase."