Upvote:9
From your refusal letter it seems you entered Britain saying you intended to stay for seven days, but actually stayed over three months. This isn't a crime, but it raises questions about how you supported yourself. The immigration officer hasn't said as much, but he believes that you were probably working illegally.
Further, the source of the additional funds provided by your family wasn't properly documented, so the officer has doubts about the funds you have available to support yourself. Again, he believes you'll probably work illegally.
Your chances of getting another UK visa in the near future are slim to none.
If you are to get another visa at all you will need to prepare a rock-solid application that addresses these points, and for this you'll need the services of a good solicitor who has experience in UK immigration.
If you have options other than the UK for your PhD you might be well advised to take them.