Microwave heating can offer rapid and homogeneous heating and thus avoids unwanted temperature gradients.
I'm confused about whether to use a singular or plural verb in the secondary clause of a sentence where the main clause has an auxiliary verb (in this case, can).
Further, how would the meaning of the sentence change if avoids was replaced with avoid?
If you mean 'microwave' as the wave in the radiation spectre, I'd recommend you such phrase as 'microwave radiation.' A listener could disambiguate the meaning of the whole sentence after such slight change. This change makes it necessary to transform the sentence into a kind as follows: Microwave radiation produces rapid and homogeneous heating in the oven and thus eliminates a possibility of unwanted temperature gradients.
If you mean 'microwave' as a microwave oven, I'd recommend such syntax as follows: Heating is rapid and homogeneous in a microwave that eliminates a possibility of unwanted temperature gradients.
If we do not resort to these changes, then the sentence is pragmatically vague. That is, it is completely not clear where it is pronounced: either at the school of chefs, or at some kind of physics exam. This uncertainty also causes the questions you asked about the syntax in your example, because people of a certain profession use usually the contexts and specific grammatical constructions inherent in this profession.
What's happening above is something similar to this:
After recovering I can walk and therefore go to the store.
Go is "borrowing" walk's subject, and it's also borrowing walk's modal.
After recovering I can walk and therefore [I can] go to the store.
The verb still has to agree with the subject even if it's "borrowed" and implied or elided.
The subject of offer in your example is microwave heating, and it's singular, so you have to use the third-person singular form avoids. Avoid is ungrammatical.
The subject of the first clause is the subject of the second clause connected by and. The verb in each of the clauses must agree in number with the number of the subject.
To make the subject clearer, you can use a pronoun:
... and thus it ....
If you wish to restate the capability:
... and thus can avoid...
or
... and thus it can avoid...
You could switch the subject to uniform heating with a relative clause, and in so doing you would avoid the tautology (i.e. homogeneous heating = no temperature gradients):
Microwave heating can offer rapid, homogeneous heating, which prevents unwanted temperature gradients.
or you can express the causal relationship:
Microwave heating can offer rapid, homogeneous heating, thereby preventing unwanted temperature gradients.
The question hits upon a particular kind of redundancy. When does microwave heat not offer “instantaneous and homogeneous heating? Never! That is what it does (strictly, one could quibble about whether the heating is strictly ‘homogeneous’.). It offers those things to anyone thinking of buying such a heater.
So there is a crude stylistic fault, which we might call all sorts of thing: unnecessary qualification (as in a cricket commentator’s: “if the bowler’s first ball had hit the wicket, the batsman would almost certainly have been out.”). But there is no grammatical error.
