'irregularly-allowed' is a compound word formed from 'irregularly' and 'allowed'. 'Irregularly' originates from Latin 'irregularis', meaning 'not regular', and 'allowed' comes from Old English 'alēowan', meaning 'to approve'.

'Irregularly' evolved from the Latin 'irregularis' through Old French 'irregulier', while 'allowed' transformed from Old English 'alēowan' to Middle English 'allowen'.

Initially, 'irregularly' meant 'not following a regular pattern', and 'allowed' meant 'to approve'. Together, they convey the idea of something being permitted in an inconsistent manner.