How to use contacts on an application in an unassuming way? Recently my former master thesis advisor (I finished more than a year ago, and want to return to academia / PhD studies after my still ongoing other research job) informed me about a summer school that fits my interests and skills very well. They suggested to mention them in the cover letter in the application process, since they are affiliated with the organizing institution, as a pro-argument and a way to possibly overcoming my recent non-academic background. 

However, I am having bad feelings about being to pushy with their name and the possibility of a resulting "I know someone, therefore I need to be admitted"-effect for the reader.

How do I benefit from my network without leaving a wrong impression? Any phrasing suggestions?
How about writing something like: "My masters thesis advisor, Prof. X, has encouraged me to apply for this summer school". No doubt you can adapt this to suit your particular circumstances.