When I finish my work contract in the UK, can I still visit a GP paid by the NHS before I leave the country? In my case it is still within the same calendar month.
Since you are an EU citizen you have a "route to settlement", meaning for you NHS services are free if you are considered an ordinary resident of the UK. This means if you have been living in the UK for the past time, especially if you have been working, then you are still a resident. The fact that you are no longer working does not matter here, as you are still a resident until you actually move out.
This only applies if you have been living in the UK while working. If you are a cross border, or posted worker it is possible that you are still a resident in your original country. In this case however you should be having your health insurance cover from your originating country and in the UK your EHIC card should be used to tell the GP that you are covered.