The headword verb is the perfect tense of ἡγεόμαι , which can be present in meaning (see H.Stephanus s.v. with several examples of it).
All the entry is taken from Photius, Lexicon eta33 (Theodoridis).
[1] Also in the perfect tense (of λογίζομαι ). According to Adler, ἥγημαι is glossed by λελόγισμαι in the Lexicon Ambrosianum. Note also that in Hesychius, Lexicon the present ἡγούμαι is glossed by λογίζομαι . Anyway, both terms are very frequently used as synonyms in all periods but especially by Patristic writers and in the Byzantine age.