“I didn't like what I saw. The fans didn't deserve that and we have to recover incredibly quickly,” Neville, who gave a flat “No” to the suggestion of quitting, told reporters at the Nou Camp.
“It's going to be a painful three or four days,” added the former Manchester United and England full back, who took charge at Valencia two months ago.
It was a cruel shock for Neville, hard on the heels of a surprise 1-0 home defeat by Sporting Gijon last weekend.
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During his time in charge, Valencia got through two rounds of the cup competition but they are still waiting for a league win and are 12th with 25 points, 26 off the pace set by Barcelona.
Levante, meanwhile, will be hoping Barca are not nearly as good as they were on Wednesday as they look for a fifth win in 23 matches to try to move off the bottom of the table.
The only glitch in Barcelona’s performance was a fourth penalty miss in eight spot kicks from Neymar, who nonchalantly took it without a run-up and hit the post.
Second placed Atletico Madrid, three points behind Barca, host Eibar on Saturday and Real Madrid, a point further back, travel to Granada on Sunday.
Real have Gareth Bale and Pepe still recovering from injury, but the Welshman may be available for the squad.
Coach Zinedine Zidane has reintroduced a practise the club used in the 1970s of playing basketball as part of the team’s training sessions.
No doubt it is good additional practise for tall players like Sergio Ramos and Cristiano Ronaldo, both noted for their heading ability.
(Writing by Rex Gowar; Editing by Mark Trevelyan)
