At the beginning of April I started a job at a company developing responsive web software in PHP. It was for an entry level position and they told me they understood that I didn't have much experience and were willing to put the time in so that I could grow into the role.
During my time at the company I somewhat felt like my immediate supervisor didn’t like me very much and didn’t want to help me learn the role, this could've simply been my own perception but I'm not sure. There were 2 or 3 times when he sent me a list of minor issues with the work I had done and each time I worked quickly to resolve them and take a brief note on what I did about each issue in case I was asked about it. Because I felt like my supervisor didn't like me I was trying to work hard and not to push his patience with me by asking too many questions and it may have just made the situation worse by making them think I was antisocial, again I’m not sure if this accurately describes how they felt.
Today the company told me that I wasn’t a good fit and they had to let me go. There reasons were that they felt they needed someone who could work more independently with PHP. Afterwords they told me that some of my work was actually quite good and was being put into production; some of my work has been in production for around a month now. They also mentioned that it wasn’t because of any personal issues. I made sure not to say anything rude or have an attitude about it but I probably left more abruptly than I should have.
Once I got home I messaged my immediate supervisor on Skype and apologized for this and told him that I just felt awkward about it and wanted to get out of there without making a scene or anything. Then I asked if it was OK to use them as a reference, he said yes and gave me his supervisors contact info.
Given this information, I feel like I may not have been given an accurate reason why they let me go. My question is about how I should proceed finding another job? Is it a good idea to use them as a reference? What should I put on my cover letter to address this? And how should I address this in future job interviews?