"despairing look" is a correct and usable phrase in written English.
It can be used to describe the expression on someone's face when they appear to be feeling desperate, hopeless, or generally dejected. For example, "John gave me a despairing look as he accepted the news of his family's tragedy."
Chris gives Paxton a despairing look.
"Defeating Hitler" and an earlier book, "God Is Back," are, in combination, a despairing look at the Israeli condition.
Munzer Hafnawi had the despairing look of a man who sensed that his remaining days on this earth might be few.
The novel cast a caustic and despairing look at a group of Mexican hotel residents who openly pursue lives of criminality that include illegal dealings, notably in drugs, for individual gain.
His entrance wasn't due until 20 minutes into the act, by which time, one way or another, the result would be known, and as I took a last despairing look at the TV screen in the dressing room he spoke.
The three stories in In a Free State (1971), which won Britain's Booker Prize, are set in various countries; Guerrillas (1975) is a despairing look at an abortive uprising on a Caribbean island; and A Bend in the River (1979) pessimistically examines the uncertain future of a newly independent state in Central Africa.
When Coriolanus is banished from Rome, I viewed the entire scene by looking at his wife's despairing face; it is the back of Brutus that proves most eloquent when Mark Antony makes his speech.
Pretty despairing feeling, isn't it?
Despite its occasionally despairing tone, Tribe is a stirring clarion call for a return to solidarity.
(Model boats floating in a model pond at Dismaland are filled with despairing-looking model migrants).
The despairing way to look at all this is that social conditions have eroded the authority, not to say tyranny, needed to create truly memorable performances.
