Recently published paper does not cite my very relevant work

That happens. Maybe they didn't do a thorough literature study before they published, or they did but before your work was available. This is more frequent than we would like to think. Note that it's generally expected from authors to have done a reasonable effort in searching for previous work, but it's not unethical per se not to reference every previous publication that could be relevant (it's also practically very hard).

Is there anything I can do regarding that already-published paper?

No, missing a previous related work is not a valid motivation to complain to the authors or editors. Just forget about it and move on.

Is there anything I can do to prevent this from happening in the future?

Yes, become so influential in your field that reviewers will know your work enough to spot when someone else submit a related work without citing yours.

Side note:

I am keen on improving my h-index and citations count.

Without going into how this might not be what you should focus on (while very good researchers tend to have a high citation count, the inverse relation is not always true), the best way to do that is to produce high-quality research, preferably on a hot research topic.

Adding to the perfectly fine answers that already have been given: The authors of the respective paper could honestly not be aware of your previous work. You could send them an email saying something like "I really appreciated your article about this and that. I myself have published in the same area, see attached paper".

It may not entirely apply to OP's question, but the study Dion et al.: "Gendered Citation Patterns across Political Science and Social Science Methodology Fields" reveals that there may, indeed, be subtle biases in which papers get considered foundational or central, and therefore garner more citations.

As the other answers noted, self-promoting, getting involved in the review process, better networking, and emailing your work to authors when you see a relevant preprint on Arxiv, all might help. In the case of inequality, one could speculate that some authors engage in these activities more than others, or that innate biases mean that other people's prejudices affect the effectiveness of said activities.

I read a piece recently (can't seem to find it on Google now) that called for removing citation limits and encouraging more comprehensive referencing, to better acknowledge the contributions of young and minority investigators. The implication was that: when the number of references is limited, authors favor works by established (often white-male) colleagues, which might be considered more "canonical", simply due to network effects.

We might be able to improve this as peer-reviewers: whenever we get a manuscript to review, we should always perform a comprehensive literature search to ensure that the authors are not unintentionally ignoring some papers—especially works by young or minority researchers, which may be high quality but neglected due to bias and network effects.