If your PSI scores have recently improved, it's worth noting that could just be down to the update rather than improvements to your site speed.
If your PSI scores have not improved, read on to find out what action you should take.
About the PSI update
Up until now, your PSI scores were fetched using HTTP/1.x due to limitations in the connectivity infrastructure of Google. This previous protocol put constraints on the amount of data that could be rendered. If you've not seen much improvement to your PSI score despite making positive changes, this could explain why.
With the new HTTP/2 taking the lead for generating PSI scores, Google is now able to render pages faster, meaning they can gather a speed score in a shorter time period. As a result of this, Google announced that "in general, performance scores across all PageSpeed Insight scores went up by a few points".
[HTTP/2] hides all the complexity from our applications within the new framing layer.
As a result, all existing applications can be delivered without modification.
As most servers support HTTP/2, it's likely you won't need to do anything except keep working to improve your PSI score. However, if you haven't noticed a change, it'll be important to contact your web host customer support to find out if their server is able to host HTTP/2.
There will likely be a lot of other factors and further improvements that your competitors — who may or may not have seen improved PSI scores — need to take into account before they may see significantly higher-ranking pages in Google.
As per the above quote from Google, those with the HTTP/2 protocol only had their PSI scores go up by a few points. In most cases, this won't have had a notable impact on search performance.
About PageSpeed Insights
As well as providing you with an analysis of how easy it is for users to access and browse your website, PageSpeed Insights are also an important factor for Google when ranking your website.
Google uses two different types of data to create PSI scores: lab data and field data.
Lab data: Tests a site using simulated devices and pinpointing from this where your website might be delivering a slower loading time for users.
Field data: Collects data in real-time as users browse the website through Chrome and identifies the performance your website visitors are getting at the time of being onsite.