system
Given the following post, respond as if you're commenting on Reddit:
human
With the introduction of vaccines, even if a variant isn’t strictly “covered” by the vaccine due to changes in proteins, would there be enough “training” of the body’s immune system on how to deal with a corona virus that the body would be more efficient if infected later?

At the start of the Pandemic, a lot of the concern was about the novelty of the virus and the fact that the majority of humans would most likely not have contacted a coronavirus and so the body wouldn’t know how to efficiently “fight” it

This is often cited as why the flu is no longer (generally) as bad, due to “people” having seen some form of flu at some point in their life.
gpt
Short answer: Yes

Long answer: Probably

When you get exposed to an antigen (by getting infected or by getting a vaccine), your body generates B cells (generate antibodies) and T cells (recognize and kill infected cells). 

I'll focus on B cells: 

First, there will be a B cell that, just by random chance, has some affinity for the antigen. That B cell will divide like gangbusters but, in doing so, will undergo something called "somatic hyper mutation". This means that the DNA that codes for the "recognition" part of the B cell will be modified in countless ways. Invariably, some of these new B cells will be even better at recognizing the new antigen, and these will be selected for. In the end, you have a population of B cells producing antibodies that have been selected for their ability to bind the novel antigen really well.

Generally, these antibodies will also be good at binding other things that *look* a lot like the first antigen so small mutations in the virus won't be able to escape.

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A (silly) analogy:

Gang of criminals from London moves into your neighborhood and your crime-fighting robot AI generates anti-gang drones. Some recognize the tattoos of Big Ben that all the members have, some recognize the Union Jack, some listen for their specific slang words ("Bloody hell!") or go after mini Coopers. The drones are, of course, tested first to make sure they don't accidentally recognize civilians (in the body this is called "negative selection").

Obviously, if a brand new Viking gang moves in and they're from Norway and drive motorcycles and wear furry horn hats, your crime fighting drones will not be much help. If, on the other hand, another gang from London moves in, your current drones might work pretty well and require few adjustments.

A bit more of a stretch might be a gang from Australia. Some of your drones might recognize just enough aspects of the new guys to keep them at bay while a new batch of drones that goes after cans of Fosters and dudes with crocodile teeth sewn into their hats comes online.