Just to clarify, immunity doesn't encourage mutations; mutations are totally random. The only impact it has is that whatever mutations do randomly occur, the ones that are capable of evading existing immunity will have a competitive advantage over ones that aren't.
But in the aggregate, existing immunity to will tend to reduce the level of mutations that do occur for two reasons:
1) it reduces the number of replications that happen vs. the same spread in an immunologically naive population. It does this both by preventing infection in the first place (as we saw pre-Omicron) and by equipping your body to neutralize the virus faster (studies have repeatedly shown that the quantity of virions is significantly lower in vaccinated/previously infected people).
2) it limits the ways in which the virus can mutate and still spread through the population. With low levels of immunity, basically anything can happen to the virus and it can have a chance becoming dominant. But with existing immunity through vaccinations and infections, certain mutations aren't going to go anywhere because they'll be neutralized. So the number of mutations that are required to give it a competitive advantage will likely be higher. And because mutations are random, that will tend to take a longer time than with low levels of immunity.
And the evidence has borne that out. Alpha, Beta, Delta, Gamma, etc. all came about in the first 100-200 million worldwide infections during the first year of the pandemic when there was little immunity. Whereas in the hundreds of millions (or even billions) of infections post-Omicron, the level of mutation is more muted, even while the number of infections is many times higher.
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