Computing statistical significance from experimental data As described in my previous question, I am currently conducting an epidemiological study in a health institution where we investigate the prevalence of Corona virus infections by measuring the existence of antibodies.
We now have the results of our study and are looking at 424 participants in a population of 785 individuals and a prevalence of 2.8% (12 individuals). As posted in my question before, I used the formula from Niang 1 to calculate the necessary sample size for a 95% confidence interval and arrived at 618.
Now to my question: Since we only have 424 participants, I guess that we will not be able to extrapolate this data for the whole population? Or is there a way to do so, after all? Thank you for your guidance!
1 NAING, L., WINN, T. & RUSLI, B. 2006. Practical issues in calculating the sample size for prevalence studies. Archives of orofacial Sciences, 1, 9-14.
 A: 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—but not that it's untrustworthy.†
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. You wouldn't generally expect the true no. cases in the hospital population to exactly coincide with your estimate; confidence intervals are a way to express quantitatively the uncertainty arising from your only having sampled a fraction of that population. For a 95%, say, confidence interval, you use the observed no. cases in the sample to calculate bounds on the unknown no. cases in the population following a procedure such that if you were to repeat the sampling many, many times, bounds thus calculated would contain the true no. cases, no matter what it is, in least 95% of these hypothetical samples.
It's worth reiterating @mkt's point: both point & interval estimates rely on the sampling scheme's being adequately modelled for their validity. If e.g. 424 names were drawn from a hat, & all those patients tested, SRS would be an entirely appropriate model; if e.g. 424 consented to the antibody test out of 618 asked, you'd have to make a case based on clinical data for the 194 patients that refused, & on their stated reason for refusal, that they were neither more nor less likely to test positive for COVID-19 antibodies than patients who consented.

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