I said that I left the last role (on the advice of my universities career advice service). I was indeed thinking about leaving but was actually informed that I had failed my probation period before it was over. This was all mainly due to suffering from depression which I have informed the potential employer of. I never explicitly lied about whether I was let go at the last role (and wouldn't) but how much effect could this have now that the potential employer is looking for references and what can I do about it?
References are not usually about your competence on the job - that has already been determined at your interview.
When requesting references, I'm looking for confirmation that they did work where they claimed on their CV, do they turn up on time, have there been any conduct issues? What are there weak areas that need supporting?
The key phrase "offer subject to satisfactory references" is typically used. What this means is that your job offer will be pulled if they are unable to obtain references, or if references uncover something incompatible with working at your new company.
I have given a fair, but positive reference for someone who was sacked, that was well received by their new employer, likewise I have refused to give a reference for someone who had claimed to work for me, when in reality, we interviewed them, and rejected them at interview - with a strong suspicion at the time that their CV was bullshit.
Generally agree with @AndreiROM not to provide reference for the position where you failed probation. Although, it's unclear just how short your time in that role was? Probation typically lasts 6mo-1yr. This may be considered adequate for a reference, especially in this case, where the only previous job was a an internship...
So this is not as clear-cut given lack of prior work experience. What is probably certain is that a reference from the position from which you were let go is not going to help your candidacy. Once we establish that, what other options are there?
Option 1: Give the internship reference only. You may have to explain why no reference from a more recent job is provided. Saying that you did not pass probation probably would be as bad as providing a reference that would say the same. The other option is to say that you do not consider time in that role sufficient for a reference to provide a fair assessment of your abilities. This may seem suspicious, but not overtly negative.
Option 2: Give the internship reference (assuming they have good things to say about your performance), AND a reference from your most recent job, for someone other than the individual who let you go. Was there someone else you worked with, who thought your performance was satisfactory? If so, tell them your thinking and ask them if they could serve as a reference. Hopefully they will understand the situation and be willing to put in a good word for you.
Option 3: Provide no references. If they press you for them, and say that references are required before your application can be considered, you can say that due to limited and brief work experience you simply do not have references at this early point in your career. Let them decide what to make of that and how to act on this. It is your right to refuse to provide any application document and to provide an explanation for your decision.
Option 4: Look for other positions, where references are not required. This may be necessary at this point since there is a chance this job might not work out due to the reference issue. Not all positions are equally bent on requiring references, and I am sure with enough perseverance you can find an employer that won't make this a sticking point.
The greater issue I see here is the depression itself. I hope you are on a path to healing, proactively tackling this condition, learning all you can about it, and seeking a long-term solution. This is not easy but this is probably the only way to ensure more stable, productive, successful career in the long term. Thank you for an honest question. Good luck!