Sorry I can't get this down to one thing we can do.
There are a number of ideas;
The first thing to do might be to add some meta tags to the SSi homepage. Google relies heavily on these to categorize/rank a website.
For example https://learnwelsh.cymru/ has the following tags, notice in particular the 'keywords';
<meta name="keywords" content="Learn Welsh, Welsh courses, Welsh classes, Welsh night classes, Welsh beginners, Welsh improvers, Welsh daytime classes, speak Welsh, practice Welsh, Welsh conversational class, Welsh language, history of Welsh, find a Welsh course, Welsh courses near me" />
<meta property="og:title" content="Croeso - welcome" />
<meta property="og:description" content="If you'd like to practice or improve your Welsh, or if you're looking for a beginners' course for a friend or relative, you're in the right place!" />
The SSi homepage appears to have no tags at all at the moment (that I can find - maybe they are hidden away!). Adding tags especially keywords WILL have a positive impact.
Add more information about the course on the homepage, and other pages to increase the 'keyword density'. This can improve your ranking. Currently the SSi landing page is beautifully sleek, but this may not be helping the ranking.
3) Google Analytics. Connect the site with this service from Google, to get info on how people are finding the site already, and what keywords you may want to target. This will make implementing ideas 1 and 2 more scientific.
Some ideas people could help out with, [NB; It's worth doing the above two ideas first]
4) Ask users to search for, 'learn welsh', or another chosen set of keyphrases. Then ask them to click through to results page 5 and click on the SSi link. If enough people do this, it could improve the site ranking as it will be seen as a popular selection for people with that query. I'm not sure how effective this would be having never done this. It also feels dodgy, and if Google found out this was happening, it may not turn out well at all.
5) Ask anyone who has a blog to link to SSi permanently. Also ask them whether they could write an article about SSi and then link to SSi.
6) Write interesting articles and get them on the SSi site, encourage users to share them, thus drive traffic to the main SSi site, and hopefully improve the ranking by increasing page visits.
7) Ask users to write articales about SSiW for news website, that can then link to SSi.
8) Get content on other platforms, especially ones that Google own. e.g. YouTube and GooglePlay. For example if the search term changes to 'learn welsh app', then SSiW comes up in 5th Place due to the GooglePlay listing.
9) Buy more domain names, which host simple one page sites that offer some info on SSiW and direct users to the main website.
10) All else fails; buy keywords/phrases from google, i.e. get in the sponsored results.