Real should have little problem seeing off third-tier Alcoyano at the Bernabeu on Tuesday, holding a 4-1 lead from the first leg of their last-32 tie.
More importantly, the competition may help to act as a diversion for the next three months.
Barca's record-equalling unbeaten start to a season, powering to 12 wins from 13 games, has piled pressure on their arch-rivals, who have slipped to three defeats in the same period, following Saturday's 1-0 upset at Real Betis.
Mourinho's side only lost two games on their way to the title last year and few, if any, Real coaches have survived to the end of a season after slipping so far behind the leaders at such an early stage.
Guiding Real to a 10th European Cup triumph remains Mourinho's primary challenge, and he has already led them to the last 16 of the Champions League.
The former Chelsea and Inter Milan boss first has to solve another dip in his side's form, which has led to questions about his tactics in four of their last five outings.
"His jokes and his excuses aren't funny any more," one columnist wrote in sports daily Marca on Monday.
"Madrid fans are still waiting for him to explain to them what is happening in footballing terms. [With him] it is all paranoia, conspiracies and fixes."
Another Madrid-based sports daily As dismissed Mourinho's explanations on Monday: "No boss, the basic problem for the defeat lies somewhere else."
Mourinho will likely rest most of his key players in preparation for next Saturday's visit of Atletico Madrid.
La Liga's second-placed side travel to the Bernabeu with a rare eight-point lead over their third-placed neighbours, and a chance to seriously rock the boat.
The King's Cup last-32 second legs are spread over the week, with holders Barcelona defending a 3-0 lead at home to third-tier Alaves on Wednesday.
Last season's finalists Athletic Bilbao have had their second leg at home to Eibar postponed until December 12 because of a rearranged Europa League fixture.
