"Scientist Training Programme" should, I think, be capitalised all the way through as it's the name of a formal scheme so I would consider it as a proper noun (phrase) I think? You also wrote "or" instead of "of" for "of radiotherapy and nuclear medicine". Definitely make sure you proofread.
Also both of these areas are clinical areas, which medical physicists work alongside doctors and other healthcare professionals in but it doesn't sound like you have looked too much into what the medical physicists do in those areas. I would recommend being more specific after doing some more research. Would probably rephrase the bit about "people who are suffering" as the phrasing seems a little less ideal. I suspect medical physicists are not going to be engaging with patients so much and would work more in the "background" of it managing the radiation safety and QA aspects, supporting the delivery of new technologies in the department to provide more care options/better care etc.
The application does specifically instruct you to include your hobbies and interests as well as plans for the future - you did not really include the former two things. You need to use examples from your hobbies and interests to illustrate transferable skills that will make you a useful member of the team in the placement e.g. "outside of my studies I was on the committee for X society as Y officer, in this role I developed Z skill which I believe will enable me to better support *whatever* in this placement" or something like that. Also your interests as phrased are a little vague and your future plans I think are also just a bit basic perhaps. You want to apply to the STP - great - what would be your long term goals after that? How will you going onto that role be beneficial to the NHS?
They're aware that you want this career and it will be helpful to you they need to know how you will be usefuf to them as well. So remember that - you need to sell yourself to them, not (just) convince them why this is great for you.