"He remains a great leader despite infirmity. "The allowance is paid regardless of income"
Why have we not used "an income" and "an infirmity" with "regardless" and "despite"?
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It's one of those quirks / general usage of English. You don't always have to use the indefinite article. It also implies a singular item which may not be the case.
In the first sentence
He remains a great leader despite an infirmity
would be read as him having one infirmity whilst in your original version he may have several.
In your second sentence it is not so clear. an income would usually be read as having any income at all whether it be one or several sources. And consequently the indefinite article can be omitted without change in meaning. From the context it appears that some kind of payment is made regardless of the recipient having any other source of income.