'monarchically' originates from English: formed from 'monarchical' + the adverbial suffix '-ly'. The root 'monarch' ultimately comes from Greek 'monárkhēs', where 'mon-' meant 'one' and 'arkhē' meant 'rule'.

'monárkhēs' (Greek) passed into Late Latin and Old French as forms related to 'monarch(a)', entered Middle English as 'monarch', then produced adjectives 'monarchic' and 'monarchical', and finally the adverb 'monarchically' was formed by adding '-ly'.

Initially it referred to 'rule by one' (a single sovereign). Over time the word broadened to describe things relating to or characteristic of a monarch, and then to an adverb meaning 'in a monarch-like/regal manner'.