You don't say the games that will be played. For better advice, let us know what these are likely to be.
Here are a couple of useful lists - click here & click here - of laptop graphics cards with how some games fare on them. The best graphics cards are at the top, with lower performance as one goes down the list. Clicking on a particular graphics card link, takes one to a page with more details, with laptop brands & models that contain them towards the bottom.
From the above, you'll find that £600 does not go very far. Unfortunately a good, or even average, dedicated graphics on a laptop is one thing that can greatly bloat its price. For £600 you're likely to get a better desktop gaming system, including a TFT monitor - so consider one of these instead. G