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Holders Hibernian ran out 3-1 winners against their city rivals in their fifth-round replay at Easter Road.
"That's quite a few backward steps and we've got a lot of making up to do to a lot of people," Cathro said.
"Maybe the mistake has been us overestimating the stage we are at as a team."
The result followed four matches unbeaten for the Premiership side but they could not extend that run against the Championship leaders.
"We lost those situations earlier in the game and caused a very hurtful evening for a lot of people," Cathro told BBC Scotland.
"Fight, second balls, 50-50s, individual things, that's what our problems were. We accept that, we feel it also. I feel it."
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Cathro sent his players out early from the half-time interval, and made two substitutes, but with his team trailing to goals from Jason Cummings and Grant Holt, the game was already lost.
The Hearts fans booed their team off at the interval and many left before the end of the game. Cathro admitted his side had regressed from recent progress and took full responsibility.
"Maybe the mistake has been us overestimating the stage we're at as a team and forgetting slightly that we needed to focus on the more simple, basic things in the game," he explained.
"One hundred percent, that's my job, that's what to do. I want to step up and do that. We overestimated where we are and we need to find solutions to it quite quickly.
"I didn't see it coming, of course not. We've not done enough on the pitch to respect the club, so we have a lot of things that we need to fix. And we will."
Hearts return to league duty on Saturday, with a trip to Partick Thistle in the Premiership before hosting Ross County next Wednesday.
Hearts players "failed to fight hard enough for the shirt" in their Edinburgh derby defeat in the Scottish Cup, says head coach Ian Cathro.