i would imagine its a transitional problem to do with something called Executive Dysfunction, and my DS15 has the same issues.
He needs some adjustments made to help give him a kickstart, and its quite a common problem, people mistake it for being lack of motivation, and it isn't... it's to do with being able to 'start' a thing, and for ANYONE with ADHD, it can be a bit of a mountain to climb to actually get things going.
Sometimes its a confidence thing (i won't be able/can't do the thing) and sometimes its that beginning simply needs to be broken down into small steps, and one of the WORST issues my DS has is moving from working on the board/talking stuff through with the teacher, to actually knowing HOW to start the work.
what my DS's TA used to do (he had 1:1) was provide him with some small prompts at the beginning of each task to give him something specific to help him move towards getting started on the work required.
The reward is to do with the provision of dopamine. ADHD is a LACK of dopamine, the 'reward' hormone, its what gives us pleasure when we do something we like/eat something we love, and we get that little kick of happiness, and its pretty vital to brain function.
When people with ADHD get a 'hit' of dopamine, it kicks the brain into gear and provides the chemical that their brain needs to fire/operate.. so rewards provide dopamine, which is why your DS is able to do the work.
The school need to tackle that Executive Dysfunction by providing some kind of dopamine hit or prompts to aid transition to initiate working for your son to be able to do his work. Breaking tasks down, providing prompts, and small rewards will do this.. something else that can also help is fidget toys which can help them focus.