Cyprus, like most countries, has private and state medicine. Please note that the services of private hospitals and clinics are expensive. Also, it costs to contact public medical institutions on a paid basis.
Since March 2019, they have launched the GESY national health care system in Cyprus, to which Cypriots contribute 6.5% of their salary. Accordingly, one can use GESY only in public medical institutions, which are much inferior in quality to private clinics.
If there is a desire and necessity, one can purchase premium-class insurance in Cyprus. It costs about 200 € per month, but it provides an extended range of medical services - non-emergency dentistry, various surgical interventions, and expensive treatment. You can choose a clinic and a doctor, including private medical institutions.