Napoli forward Piotr Zielinski put the visitors ahead five minutes before half-time, with Jorginho earning a share of the spoils in the 78th minute from the penalty spot after Jakub Blaszczykowski brought down substitute Federico Chiesa.
Mancini is charged with masterminding Italy's recovery from the humiliation of failing to qualify for the 2018 World Cup.
However, a protracted saga over his future with the Ligue 1 club after a mooted move to Marseille failed to materialise has left the 28-year-old short of match practice – something that showed over the course of an ineffective 62 minutes in Bologna.
"Mario needs to play, he is an experienced player on the international stage," Mancini told Rai Sport after Balotelli applied ice to his thigh on the bench having made way.
"But his fitness levels are an issue right now. Only the best condition is needed."
Jorginho was culpable with a misplaced pass before being dispossessed in the build-up to Poland's opener and mistakes were a recurring theme of a fitful showing.
"It's the first important game, so a few errors can happen, but overall the boys have done well," Mancini said.
"We made a few too many mistakes when playing out from the back and that allowed Poland, to use the counter-attack, which was what they wanted to do anyway."