As far as hard research evidence is concerned, it depends what kinds of cysts we're talking about here. There is, for example, some reasonable evidence that acupuncture may benefit polycystic ovarian disease or syndrome (PCOS) but not a great deal of evidence for common or garden cysts on the skin, mainly because research usually involves specific named conditions, and cysts tend to vary with location and cause.
However, from a Chinese medicine perspective cysts are usually regarded as accumulations of fluid which have gathered because of a local blockage or the thickening and gathering of fluids because of a failure of the overall function for even distribution of fluids in the body. Whichever it is, or even if it is both, there are a number of protocols which might be used to address the problem. The symptoms which the patients describe and the signs with which they present are the same, whatever system of diagnosis is used. Chinese medicine has worked with the same raw material for over two thousand years, and has clearly addressed problems like this since it was first formulated.
We tend to say the same thing with every answer, which is to visit a BAcC member local to you for advice and guidance on whether acupuncture is a good option for you, or whether they think other modalities might suit your case better. Most are happy to give a little time without charge to look at your unique presentation and see what Chinese medicine can offer, and equally happy to refer elsewhere if in their judgement there are more suitable options. In Chinese medicine each patient's balance and constitution is unique; trying to give a 'one size fits all answer' clearly has limitations. But we do our best!