She begged for mercy with tears streaming down her cheeks.

Literal

She [topic-は] tears [object-を] cheeks [on-に] letting-stream [while-ながら] mercy [object-を] begged.

A dramatic, literary sentence. 伝わせる is the causative of 伝わる ('to travel along a surface'), meaning 'to let [tears] run down.' 情けを請う (なさけをこう) is archaic phrasing for 'to beg for mercy/pity' — 情け means 'compassion, mercy' and 請う means 'to beg, entreat.'