The usual issue here is that if the flight was planned for X fuel, then X is what’s required, period. And X-400 is illegal. Even though as noted, 400# of fuel is about 4 minutes of cruise burn in a 737 NG, and more like 5 or 6 minutes cruise burn in a MAX. This slightly-short-of-fuel situation happens a lot when there’s a plane change. Of course, the earlier the plane change is known, the better the odds HQ can get the revised fuel load or a top-off request to the fuelling contractor in time for them to get their task done to the correct jet before it becomes the pacing item for departure.
In general, if faced with a mere 400# shortfall on a 737-ish airplane the Captain and dispatcher will confer by phone and see if there is any slack or “just in case” fuel anywhere in the plan that can be revised out of existence. Thereby converting X-400 into a legal load. Sometimes that slack is built-in and can be prudently sacrificed in the name of a (more) on-time departure. If that is the case, the Dispatcher can datalink amended paperwork to the plane and pilots and after suitable review the Captain formally legally approves the revised plan which makes X-400 legal, the gate agent closes the door, and they push off the gate.
Sometimes there really isn’t any slack. Further, there are times when a prudent Captain would prefer another couple thousand pounds for “just in case”, but the weights, airport(s), diversion option(s), and weather don’t permit that luxury. Now’s where you earn your money. Do I push the issue and leave people behind, and incur delays for de-boarding, removing luggage or cargo, and for loading the additional top-off fuel, or do I go as planned, recognizing that although we’re 100% legal, our margins for bad luck are small and we’ll have to be extra pro-active about literally everything until we’re at the gate at the destination. And whatever our raw odds of a diversion were today, the extra-tight fuel bumps those odds by 5x. Or 20x. But we’re in the business of a) abiding 100% by the regs 100% of the time, and b) insofar as possible, moving as close to everything as close to every flight as close to on-time as possible.
With the occasional miscue that a diversion represents. Which is not to say that a diversion is proof-positive of a mistake or of corner-cutting gone bad. Sometimes weather or [whatever] just scrambles everything beyond recovery given the statistically sensible fuel planning and then executing the contingency plan of diverting is simply the smart, prudent, and, from our POV, very routine move.
You didn't ask, but ...
The converse situation is switching to a plane that was fueled for a different mission and has too much fuel for your plan. If the flight isn’t already skoshy (technical term) on weight, we just go with the excess & waste some of it burning extra to carry the useless excess weight to the next station.
But if we’re right up against one of the weight limits for takeoff, cruise performance, landing, or diversion, then we’re faced with either offloading fuel or leaving people or luggage or cargo behind. Often time becomes the decider: which of those 4 offloads (or not-yet-loadeds) can we accomplish quickest with the fewest opportunities for miscommunication amongst all the moving parts? OK, let’s do that one.