When I was faculty at an APA internship, we were looking for interns with some combination (ideally all three) of experience in a similar setting, experience with a similar population, and experience with the type of clinical services/methodology. Letters, essays, etc., were only important in so much as a) they conveyed information as to the above (though this type of info could typically be found in a better format elsewhere in the application materials); and b) conveyed any negative information about the candidate (which was very rare). I can't ever remember a case where someone without appropriate experience ever got an offer because of really good letters of recommendation. There were some where the candidate had a mentor who reached out to us personally (usually we had some prior relationship with that mentor) to let us know that the candidate would be a great fit despite their experience. There were a very small number of candidates who had appropriate experience, but due to something funky in a letter or essay did not get ranked. Negative letters were exceedingly rare- maybe saw one or two over my 5 years, and these were for candidates who were grossly under-qualified anyways. There were a few mentors from schools that often sent us candidates whose letter we say a lot, and you learned to pick up on some relative language (e.g., "top 5% of students I've ever had" vs. "top 10% of students I ever had"), but otherwise not very useful. Biggest issue in essays was when they espoused training or career goals that we couldn't help out with (hint: if you are applying to a program that does primarily ABA, with just a touch of CBT in a minor rotation, and works with children, don't state that your career goal is to apply psychodynamic theory in your work with adults).
