something like:
Dear (insert name of admissions tutor for psychology)
I am writing with regards to my application for a place on the BSc Psychology couse starting september 2004. Unfortunately, my application was unsucessful. Under the circumstances this was not surprising, as I had underperformed in my AS exams last year, and so was only predicted BCC, which obviously is much lower than your usual offers. However, I retook many of my AS exams in January of this year, as I knew that I could improve them and do better. I managed to significantly improve my marks and grades in these exams. Although officially my school has not changed my grade preditiction, as it is not customary for them to do this, I am certain that I have put the work in to achieve the necessary grades for your course. I am aware that you may have a few places still avaliable and would very much like to be considered for one. I have very strong ambitions to pursue a Psychology degree as............(insert reasons why you want to do a psychology degree) ultimiately leading to ....(insert jobs you would like to do in the future). I would particularly like to study at Manchester becuase....(insert reasons, perhaps including reputation, been to look round and liked the atmosphere etc...)
If it is not possible for me to be admitted this year, perhaps you would consider me for a place commencing september 2005? If not, then I will reapply to you next year, as I am confident that I have the qualities and skills needed to study Psychology at Manchester, and I would really appreciate the opportunity to pursure my dream.
Yours sincerely,
(insert your name)
(insert contact details, name address,telephone number, ucas number etc.)