Mancini took over from Walter Mazzarri last month, but has taken just one point from three Serie A matches so far.
Inter did defeat Dnipro 2-1 in the UEFA Europa League in Mancini's second game in charge, but Sunday's 2-1 home reverse at the hands of Udinese left the head coach fuming.
Having led at San Siro, Inter allowed their opponents back into the contest and ultimately fell to defeat when Rodrigo Palacio's poor backpass let in Cyril Thereau for the winner.
Mancini won three successive Scudetti in his first stint at the helm but with Inter currently lying 12th in the table, Kovacic says patience is needed.
"A coach can't change everything in a heartbeat," he told the club's official website.
"Unfortunately, we're in one of those periods and we just need to look forward. We've got a good team but we're making too many errors.
"We must listen to the coach and work harder.
"The most important thing is to do well wherever the boss asks me to play. Is there too much responsibility on me? No, I just concentrate on my game. I want to help the team.
"We're not having a good spell at the moment, but let's give Mancini time.
"Serie A is a very tough league, and every match is hard. We need to try and do better, one good half is just not enough."
Next up for Inter is Thursday's Europa League trip to Qarabag, with their next domestic assigment coming in next Monday's league visit to struggling Chievo.