Is the Hadith: “Gabriel is entrusted with the needs of the Children of Adam. If a disbelieving servant supplicates, Allah Almighty says: 'O Gabriel, fulfill his need, for I dislike to hear his supplication.' And if a believing servant supplicates, Allah Almighty says: 'O Gabriel, withhold his need, for I love to hear his supplication,'” authentic?

Allah Almighty promised to answer the prayers of the believers, as stated in His words: "And when My servants ask you concerning Me, then I am indeed near. I respond to the invocation of the supplicant when he calls upon Me," and His saying: "Call upon Me; I will respond to you." The prayer of a Muslim is answered as long as he does not pray for a sin or for the severance of kinship ties, and as long as he does not become impatient, thinking that his prayer has not been answered, and then abandons supplication. Allah may choose for His servant something better than what he asked for, storing it for him in Paradise, or averting affliction from him because of that supplication. There is no Muslim who supplicates with a supplication except that Allah grants him one of three things: either He hastens his supplication for him, or He stores it for him in the Hereafter, or He averts from him an equivalent amount of evil. Indeed, Allah becomes angry with those who do not ask Him. It is advisable to seek prompt answers by supplicating during times when prayers are answered, by supplicating with Allah's Greatest Name (Al-Ism Al-A'zam), with the supplication of Yunus, and by having high aspirations in supplication. As for the hadith mentioned in the question, it is not authentically preserved, and what is preserved is its attribution to Thabit.