To besmirch or sully someone's reputation; to disgrace or slander someone (idiomatic).
彼はライバルの評判に泥を塗るようなうわさを流した。
He spread rumors that tarnished his rival's reputation.
試合での不祥事がチームの名に泥を塗った。
The scandal in the match sullied the team's name.
無責任な発言で先輩に泥を塗るのはやめなさい。
Stop making irresponsible remarks that would disgrace your senior.