Høegh formerly played for SC Heerenveen, FC Basel, Odense Boldklub and Midtjylland.
Born in Odense, Denmark, Høegh started played for OB's youth ranks from 2004 to 2010.
[4] Høegh returned to the squad in March 2016 and the first game that he played, coming in as a substitute, was in the 2015–16 UEFA Europa League knockout phase, in the round of 16, at home against Sevilla.
[11] On 5 October 2020, Høegh joined Danish Superliga champions FC Midtjylland on a free transfer.
[12] After two years at Midtjylland, Høegh signed a three-year deal with Randers FC on 19 July 2022.