。
She coaxed him out of his bad mood.
Literal
She [topic-は] him [object-を] coaxed-and, mood [object-を] made-fix.
宥めすかす (なだめすかす) is a 複合動詞 combining 宥める ('to soothe, calm') + すかす ('to cajole, coax, distract') — captures the act of placating someone by gentle handling and small flatteries, often used with children or sulking adults. 機嫌を直す ('to recover one's mood') is a set phrase, and 直させる is its causative ('make X recover their mood'). The te-form 宥めすかして sets up the means by which the second clause's effect was achieved — 'by coaxing him, she got him to cheer up.'