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This person can receive their 2nd dose of vaccine, if the first dose was an mRNA vaccine. However, at the time of the 2nd dose the person should be recovered from their illness. If the 1st dose of vaccine was the J&J vaccine, then no further doses of vaccine are necessary.
The CDC’s language says this: Vaccination of people with known current SARS-CoV-2 infection should be deferred until the person has recovered from the acute illness (if the person had symptoms) and they have met criteria to discontinue isolation. This recommendation applies to people who experience SARS-CoV-2 infection before receiving any vaccine dose and those who experience SARS-CoV-2 infection after the first dose of an mRNA vaccine but before receipt of the second dose.
https://www.cdc.gov/vaccines/covid-19/info-by-product/clinical-considerations.html