to OP
this is kind of a misconception amongst some people...
By law, a pilot has to plan for a certain amount of fuel on board after arriving at the destination. This is enough to make some number of attempts and fly to the planned 'alternate' destination (alternate is 'in case of weather', so a safe alternate is often in a different area with different weather patterns, rather than the 'closest' airport). When the amount of fuel is enough to fly to the alternate and land, then attempts to land at the first airport must cease.
Companies can also specify limits to the number of times a pilot can attempt to land, there are lots of reasons for a company specifying a number, it may believe that after 2 attempts the pilots will be trying too hard on the 3rd attempt, or it may be that they have researched and find it cost effective, or or or.
If a pilot thinks the plane is 'low on fuel' he can state that to the controller and usually get preferential treatment, short time in a holding pattern, etc. But when a pilot declares that the plane is low on fuel, the pilot can be asked to explain, and risks action if it turns out that the legal requirements were not met. And in some cases in the past, the airline has been asked to explain why their planes have declared low fuel.
So a long haul flight should arrive at it's destination with the same amount of 'extra' fuel on board as does a short haul flight. The distance of the flight does not reduce the requirement. In fact, (depending on the company), it is more likely that the flight arriving from a long haul will have more 'reserve' fuel than one arriving from a short flight.
Planes don't drive to the pump and fill up like people do with a car, rather they take into account the weight of the passengers and cargo, length of the flight, and determine how much fuel to carry that will complete the flight with the required (company and law) reserves.