How long after you work for a place can you ask for reference

I am working part-time (very very part-time, 10 hrs/week) for this home health agency. I have been working for them for less than a month. Can I ask them for reference while I am applying for another part time or full time job? I don't have plans to quit my current job even though the hours are not enough to pay the bill. The clients are easy to take care and the people are supportive and reasonable. If you were my employer how do you feel about this? Also, can I put down my current job as my work experience on my resume? I don't mind if the hiring manager of the new job contact my current employer because I have been a very good worker. My clients like me.