Liverpool, Bournemouth, West Bromwich Albion, Stoke City, Watford and Hull City have lost to lower-league teams this season and there are two non-league sides in round five for the first time ever.
United visit second-tier Blackburn Rovers on Sunday and having been stung by an FA Cup defeat at Newcastle United during his first season at Chelsea, Mourinho is determined not to take unnecessary risks.
"Maybe we don't have as many English managers with that culture as we should," he told reporters following United's 3-0 Europa League win over Saint-Etienne on Thursday.
"And I lost with Chelsea against a League One team (Bradford City, in January 2015). But I never threw it away. We lost because we lost. So I try always to go serious. I like Wembley, I like the FA Cup, so I have to try to get the second one."
Of the managers to have fallen to lower-league opposition in the FA Cup this season, only Liverpool's Jurgen Klopp, Watford's Walter Mazzarri and Hull's Marco Silva are not British.
Mourinho's FA Cup tie against Newcastle with Chelsea in 2005 arrived in similar circumstances, prior to a Champions League trip to Barcelona and a League Cup final meeting with Liverpool.
- 'Respect the competition' -
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Mourinho saw a weakened team lose 1-0 at St James' Park and although Chelsea overcame Barcelona and beat Liverpool in the League Cup final to give him his first English trophy, he still regrets the defeat.
"I focused too much on Barcelona and Liverpool. It was good because we beat Barcelona and we won the final against Liverpool. But the feeling I threw it away was not good.
"So I don't throw it away. If I lose, I lose because the opponent was better or because we didn't play well. But I'm not going to throw it away. I go to Blackburn with that respect. We played already Reading and Wigan from the Championship, so it's not new for us, the level and the difficulty of the Championship."
"But I'm going with a good team because I respect the competition a lot and Man United demands that you go serious to every game."
Mourinho said he did not know whether captain Wayne Rooney, midfielder Michael Carrick or playmaker Henrikh Mkhitaryan would make the trip to Ewood Park.
Rooney and Carrick missed the win over Saint-Etienne due to what Mourinho termed "minor" muscular problems, while Mkhitaryan was absent through illness.
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