Humiliated, as the one-word headline in Merseyside's football pink put it, by the side 22nd in the First Division, they visit Waddle's spiritual home on Wednesday knowing that defeat by Newcastle would leave them closer to the relegation zone than to a place in Europe. Royle has been manager a mere 26 months, but a sixth successive Premiership defeat might push the Everton board's tolerance to the limit.
The natives are certainly restless, and booed their team off before rising to Waddle. Embarrassingly for a club with their self-image, Everton's tactical and technical limitations were exposed by City much as they were by Port Vale and York in their other cup calamities of the past 12 months.
In theory the pounds 25m spent since Royle succeeded Mike Walker has improved the squad considerably; in practice Everton often look wretchedly one- dimensional. They seldom resist the temptation to aim for the head of Duncan Ferguson, whose confidence is so low that he is laying off balls rather than going for goal when in heading range.
This aerial emphasis has had the effect of marginalising Nick Barmby, while their other potential match-winner, Andrei Kanchelskis, has seemingly forgotten how to take on a defender for pace. Responsibility for crossing often fell to Earl Barrett, a full-back once famous for his reluctance to venture into the opposition half. As a result Everton rarely had the width to stretch City's defence.
Throw in the dearth of midfield guile, a problem put into sharp relief by the wiles of Waddle, and Everton can look painfully prosaic. Neville Southall, defending Royle in his local-newspaper column, felt they needed to score "off a player's backside" to spark a winning run. Their goals here were scarcely more stylish, and supporters steeped in sophisticated football were singularly unimpressed.
Hence their appreciation, tinged with anguish, of Waddle. Now 36, and heavier and slower than when Graham Taylor picked Carlton Palmer and Geoff Thomas ahead of him for England, the Geordie enigma still shambles around with the posture of Harry Enfield's alienated teenager "Kevin". But the speed of thought, the vision and, in one exquisite cameo, the sheer sense of fantasy, set him apart from the mere mortals.
One irony is that City's manager, Chris Kamara, spent much of his two decades as a "ball-winner" stifling such skills by fair means and foul. Another is that City hardly had to fight off the competition when they rescued Waddle from Falkirk in the autumn, an anomaly which Kamara attributes to the job-security fears of his Premiership counterparts.
While City's recent record of 10 points from 30 indicates that Waddle has trouble sustaining his influence, he was clearly inspired by the setting on Saturday. After striking the woodwork with a free-kick, and spraying 30-yard reverse passes around, he created the opening goal for John Dreyer and another for the strong-running Swede, Robert Steiner.
The piece de resistance, however, was the middle goal in a burst of three in 10 minutes by City. Kanchelskis, inexplicably the last line of defence, gifted possession to Waddle, whose instinctive chip from nearly 40 yards caught Southall off his line. Perfection, and from a man still horrified by the thought of the free shot from a third of that distance otherwise known as the penalty kick.
Everton twice cut the deficit without seriously threatening a replay. Royle anticipated a "sleepless night" and may need an urgent upturn in results to survive. Bizarre as the prospect appears now, Kamara, who claims City are "the best footballing side in our division", could be under pressure himself unless they pick up the points to back up his bravado.
In the meantime, as Bradfordians guzzle the heady brew that is FA Cup glory, Everton's task is to prevent one Newcastle exhibition being followed by another.
Goals: Dreyer (49) 0-1; Waddle (51) 0-2; O'Brien og (54) 1-2; Steiner (59) 1-3; Speed (90) 2-3.
Everton (4-4-1-1): Southall; Barrett, Watson, Short, Phelan (Grant, 58); Kanchelskis, Parkinson, Stuart, Speed; Barmby; Ferguson. Substitutes not used: Unsworth, Gerrard (gk).
Bradford City (3-1-5-1): Schwarzer; O'Brien, Mohan, Sas; Dreyer; Hamilton (Liburd, h/t), Duxbury, Waddle, Kiwomya (Stallard, h/t), Jacobs; Steiner. Substitute not used: Sergio Pinto.
Referee: M Reed (Birmingham).
Bookings: Everton: Stuart, Speed. City: Steiner, O'Brien, Duxbury.
Man of the match: Waddle.
Attendance: 30,077.Reuse content