Hi everyone. I'm not experienced in this area and applying for a PhD works a bit different in my country so I hope I can get some help here. I'm about to apply for a PhD studentship in UK and I have a problem with writing a covering letter. My concern is that the Professor who is going to be a supervisor was my supervisor in the past when I was doing my Erasmus exchange year. That means he more or less knows me and my background. Should I write the letter as I don't know the person I am writing to and describe in detail the work I did under his supervision? Should I mention his name in the letter as my supervisor in the past? Do people from admissions read it first or is it forwarded directly to a supervisor? Thanks for any help

Definitely customize your letter for the supervisor. Introduce yourself in the opening paragraph as a former student of his during an Erasmus exchange year, and explain how during that year you developed a keen interest in his research, etc. Generally you send the letter directly to the prospective supervisor.