äşgäk is directly attested in Old Turkic texts. The reduced variant äşäk is already early: loss of the medial velar reflects the fricative pronunciation of g; Dīwān Luġāt al-Turk also has äşyäk, showing g > y. Oghuz reflexes without a medial consonant nevertheless support original g, not k.
The etymology is disputed. Clauson tentatively derives äşgäk from äş- “to amble” as a deverbal noun, while Räsänen derives it from *eş “companion” plus diminutive -(g)äk. Starostin–Dybo–Mudrak compare it with Mongolic *elǯigen and Tungusic *eji-ken within their disputed Altaic reconstruction *eĺčgek > eşşgek. Ünal (2022) instead proposes *ät₂gä “donkey”, derived from *ät₂i “ear” with an original sense ‘eared (animal)’; this newer proposal remains hypothetical.