The work deals with laws and customs, particularly relating to the holidays. It is strongly associated with Sabbatianism and was attributed by R. Yaakov Emden (e.g. in his Torat HaK'naot, Altona, 1752, as noted here) to Nathan of Gaza, Shabbetai Tsvi's prophet. Many other rabbis followed R. Emden's assessment. Although not all of its opponents thought it was necessarily written by Nathan of Gaza, it was thought to be Sabbatian by many rabbis, both Sephardi and Ashkenazi.
Some rabbis, however, endorsed the work. R. Menahem Heilprin, for example, wrote this whole work K'vod Hakhamim to defend the book.
Note also this article which notes that the author of Yesod V'Shoresh HaAvodah in his Last Will and Testament praises the study of Hemdat Yamim. In at least one edition of his will, the reference to the Hemdat Yamim was removed.
Similarly, as noted, the Hayyei Adam (klal 144) quotes the Hemdat Yamim as the source for Tefillat Zakka and others, especially Sephardi writers, quote the work regularly.
For example, as noted here, R. Hayyim Palache in Kol ha-Hayyim (Ma’arekhet he, no. 18, pp. 17 ff.) defends the Hemdat Yamim, claiming that we should not disqualify it, no matter what the author believed about Shabbetai Tsvi.
As noted here, historians have similarly debated the nature of this work. Boaz Huss, following Yeshaya Tishby and Gershom Scholem, believed the work to be Sabbatian, while Avraham Yaari was of the opinion that it was not Sabbatian. As noted here, David Kohn also held the work to be Nathan of Gaza's.
For R. Yehiel Goldhaber's lengthy discussion of the entire debate, see here.