The Black Stars lost 2-1 to the minnows on Sunday and the loss was Hughton's fourth defeat in 11 games in charge.
Some reports suggested Hughton, 65, had been assaulted by a fan when the team returned to their hotel but Ghana said it was only a "verbal altercation".
"We've dealt with the issue internally," a Ghana spokesperson said.
Former Nottingham Forest, Brighton and Newcastle manager Hughton became Ghana coach in February 2022, having previously been a technical advisor in the Black Stars' set-up.
Four-time Afcon champions Ghana were unbeaten in their first five games under the former Tottenham full-back but have lost four of their past six.
Two of those defeats have been significant upsets, the loss to Cape Verde following on from a World Cup qualification shock at Comoros in November.
"We have had a meeting with the coach and gave him 100% assurance that we are solidly behind him," Ghana FA's spokesperson Henry Asante-Twum added to BBC Sport.
"We think it is possible for the team to win the two remaining matches to make it to the next round."
Asante-Twum said that security at Ghana's hotel had been "beefed up" after the incident that involved a single supporter who confronted Hughton about the team's under-performance.
"There was only a verbal confrontation," he added. "He was actually trying to get closer to the coach and then the security on duty intervened - there was no arrest but security took him away from the scene.
"Normally after games, tempers are high, emotions go up - we will try as much as possible to not allow this thing to repeat itself."