Flustered by a careless mistake, she burst into tears.
Literal
She [topic-は] careless mistake [at-に] flustered, wah [and-と] started-crying.
まごつく ('to be flustered/confused') is an expressive verb. わっと is an onomatopoeia for a sudden burst of crying — literally the sound of wailing. 泣き出す uses ~出す for sudden onset.