You've calculated the minimum sample size necessary to achieve a desired level of precision in your estimate of population prevalence. Though your initial guesses as to the hospital population & prevalence of COVID-19 infection don't matter now you have data, your sample size calculation does assume simple random sampling (without replacement): if that assumption remains valid, you can calculate confidence intervals (or perform hypothesis tests) as planned from the true population size & no. cases observed. That you've ended up taking a sample smaller than planned means the  confidence interval may well be wider than you wanted&mdash;but not that it's untrustworthy.<sup>&dagger;</sup>

You seem to be thinking in terms of a point estimate's being valid ("extrapolatable to the population") when the sample size exceeds a certain value, else not; which isn't the right way to think about it.

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&dagger; You may have more reason to use exact methods rather than an asymptotic approximation, however.