Hyperemesis gravidarum mentioned in the title of this post refers to "prolonged and severe nausea and vomiting" during pregnancy alongside other symptoms. It's thought to affect only a relatively small percentage of pregnant women despite vomiting and nausea being pretty common throughout many pregnancies. Indeed, hyperemesis gravidarum (HG) is much more than just nausea and vomiting.
The findings reported by Marlena Fejzo and colleagues  looked at the "neurodevelopmental outcomes of 267 children delivered by 177 mothers with HG were compared to neurodevelopmental outcomes from 93 children delivered by 60 unaffected mothers." The study was undertaken on the basis that previous published research from this study group  had identified "neurodevelopmental delay" as a possible *correlate* tied to HG exposure during pregnancy. That being said, on that last occasion researchers also described finding "no evidence for increased risk of 13 emotional, behavioral, and learning disorders, including autism, intellectual impairment, and obsessive-compulsive disorder" in their cohort.
This time around: "Similar to at age 8, the children (now 12) exposed in utero to HG had over 3-fold increase in odds of neurodevelopmental disorders including attention, anxiety, sensory, sleep difficulty, and social development delay/social anxiety." With a longer follow-up period, researchers also reversed their 'no evidence for increased risk' sentiments with regards to a diagnosis of autism in their cohort. So: "there was also a significant increase in Autism Spectrum Disorder (ASD), reported in 22/267 (8%) of children exposed to HG in utero and no unexposed children."
Let me be clear on this: these latest results on their own don't necessarily mean that HG *causes* offspring autism. Although 8% of the HG exposed group reporting autism or ASD seems quite high, it has to be seen in light of an increasing (estimated) prevalence of autism more generally (see here and see here). I'd also suggest that as with many other pregnancy/gestational factors put forward as potentially influencing offspring autism risk, one needs to be mindful of the issues involved in teasing out which is the important variable from potentially many (see here and see here).
But... further investigation is indicated in this area. Indeed, I hark back to the findings reported by Andrew Whitehouse and colleagues  who observed a "strong, positive association between increasing frequency and severity of NVP [nausea and vomiting during pregnancy] and ASD severity in offspring." No, not necessarily HG, but an implication that some of the overt features of HG might hold some *connection* to offspring autism risk for whatever reasons...
 Fejzo M. et al. Analysis of neurodevelopmental delay in children exposed in utero to hyperemesis gravidarum reveals increased reporting of autism spectrum disorder. Reprod Toxicol. 2018 Dec 27. pii: S0890-6238(18)30558-6.
 Fejzo MS. et al. Neurodevelopmental delay in children exposed in utero to hyperemesis gravidarum. Eur J Obstet Gynecol Reprod Biol. 2015 Jun;189:79-84.
 Whitehouse AJO. et al. Symptom severity in autism spectrum disorder is related to the frequency and severity of nausea and vomiting during pregnancy: a retrospective case-control study. Mol Autism. 2018 Jun 19;9:37.