Even though it is difficult to break into the top four, at least Manchester United has revived that hope. Being six points behind Tottenham can be said to be a bit difficult to overcome, but if the Red Devils' performance continues to improve then it is not impossible for United to take over. Apart from that, if we look at Manchester United's remaining schedule, it looks like the obstacles will be tougher to overcome, Erik Ten Hag must prepare his troops to overcome tough obstacles from strong teams such as Manchester City, Arsenal and Liverpool.
At least they are doing their jobs well that is enough to give them hope that they can achieve a position in top four at the end. It's certainly that Tottenham and Aston Villa will lose points in rest of season and will give opportunities to Manchester United. It means if Manchester United can take advantage of bad matches from two competitors, they are able to get a fourth position. They will have to play well and get good results from rest matches, from strong to average and weak opponents.
I have to say that with good performance from Manchester United it is not enough because they also have to expect Tottenham's bad performance whatever Manchester United's efforts will be in vain if Tottenham is in the same situation so to make it happen both teams must be in different performance, no other way. In goal productivity, Manchester United are far behind Tottenham, Tottenham with 15 goals differences and Manchester United 0 scored 33 goals conceded 33 goals that's why I say it's so difficult for Manchester United to pursue Tottengam.
It's not enough for now but it's better than nothing or if Manchester United have poor performances. They are creating opportunities for themselves by winning recent matches. With 6 and 5 point differences against Aston Villa and Tottenham Hotspur, Manchester United will see bigger chance if two competitors lost one of their matches as it will turn differences to only 2 and 3 points.
You are right that Manchester United will have to aim at having more total points than Tottenham Hotspur because it's hard for them to compare in goal differences if two clubs have same total points.