I downloaded the paper and the accompanying data in order to carry out my own analysis. As many people have already pointed out the study is a particularly poor design and does not meet even its own power requirements of 48 participants. It is not a randomised trial and it introduces confounding variables such a participant age, and it is not blinded so the placebo effect is also an issue. Elizabeth Bik also pointed out that the final measure is unreliable and patients go from being negative to positive because of a dubious threshold.
My main concern is the very large number of non-determined data-points in the dataset. I copied the dataset into SPSS as a binary set replacing the significant test measures with positive and the negative test results as negative. Non-determined values were marked as missing values as were undeterminable values such as time since symptoms developed for someone who was unsymptomatic.
I then carried out the Fishers Exact Tests for those taking the hydroxychloroquine and those left untreated for days 1 to 6. I could not reproduce ANY of the reported p-values. The results from my analysis are given below, but the key point is that there is no significant effect of hydroxychloroquine on the virus except for day 6 and by then because some of the participants had experienced symptoms for up to 10 days already you would have expected them to recover from the virus anyway. I sent my results to the corresponding author. My view as an editor for another journal is that if it had been soundly peer reviewed and the quality of its statistics assessed then it would never have been published.
This does not mean that hydroxychloroquine is definitely not a potential treatment. It may prove to be a treatment in a future study but this particular study presents no evidence that it is an effective treatment for Covid-19.