After enjoying a 20-match unbeaten run at the start of his Barca tenure, Martino's bubble burst with defeat to Ajax in the Champions League before an even more damaging reverse against Athletic Bilbao in La Liga at the weekend.
The defeats have cast a rare mood of gloom over the Nou Camp but Martino believes spirits can be raised by a strong showing in the cup, with his players "full of optimism" heading into Friday's fourth-round tie.
"It's a vital match," Martino said ahead of Barcelona's first encounter with Cartagena since 1989.
"A win will help us to recover emotionally.
"We know we play a team from the third category, but we have to do our job and win."
With tougher assignments on the horizon, Martino was asked if he would field a significantly weakened side for Friday's away leg.
"Remember that before the game with Ajax people spoke that perhaps it did not have much importance," he replied.
"I think we should not look beyond the next game, and the next game is Cartagena."
Barca midfielder Xavi, meanwhile, has hit out at some of the negativity that has been directed at the club in the wake of their recent struggles.
"We believe in what we are doing, but there is a negative element surrounding the club that doesn't help," Xavi, who will miss the match with a sore calf, told Catalunya Radio.
"I hear things, and what I don't hear people tell me about, and it seems there are more negative comments than even we are used to. But to be honest, the only time the team has given the impression of not being in control was in the first half against Ajax, and perhaps at times in Bilbao."
Xavi also defended Martino, saying the coach is "a leader who's really fair, knows loads about football and has very clear ideas".
Barca's game with Cartagena is one of six last-32 first legs to be held on Friday, with the other ties seeing Lleida host Real Betis, Villarreal welcome Elche, Algeciras entertain Real Sociedad, Sevilla visit Racing Santander and Real Valladolid play host to Rayo Vallecano.
Holders Atletico Madrid, who are level on points with Barca at the top of La Liga, kick off the defence of their crown on Saturday against Segunda Division B side Sant Andreu.
Atletico earned the capital bragging rights a few months ago by defeating Real Madrid 2-1 in last season's cup final at the Bernabeu, and full-back Juanfran says his team will be going all out to retain the silverware come the final on April 19.
"We're really looking forward to Saturday, starting the cup as the defending champions," he said.
"We're really enthusiastic, wanting to start, have a good game and trying to go as far as possible, and it all starts with winning against Sant Andreu."
Last year's beaten finalists are also in action against third-tier opposition on Saturday, as Carlo Ancelotti's Madrid face Olimpic at La Murta.
Elsewhere on Saturday, Getafe are the visitors to Girona, Celta Vigo play Athletic Bilbao and Recreativo Huelva host Levante.
The action wraps up on Sunday with five further ties: Alcorcon v Granada, Gimnastic de Tarragona v Valencia, Las Palmas v Almeria, Real Jaen v Espanyol and Malaga v Osasuna.
The return legs will be played between December 17-19.