Peer-review -- Can I ask to cite my already published paper that is relevant?

If you are absolutely certain that you know (say) two papers which are clearly relevant for their work, you should state it:

The authors should comment on the work presented in [1, 2] and how their work improves on that or is different.

Add more details if you want to be helpful to the authors; for example, if the work [1, 2] uses a very different methodology than the present paper, you may want to offer a bit more detail to help the authors out. But it is voluntary. Also, whatever you write might end up copy-pasted in the final paper.

If you are not absolutely sure the papers are the most relevant and would improve the paper, you might still write:

The authors may want to comment the related work [1, 2].

maybe with more qualifications or detail. You can certainly make friendly suggestions, but given the power relations in play, you should explicitly mark them as voluntary improvements or such.

Also consider that there might be a lot of related literature that would be as appropriate to discuss as yours and the authors had to cut off their literature review at some point. In such a case, asking them to cite your paper in particular seems suspect. You should adjust the certainty of your recommendations according to how well you know the field.

Well, of course there is some conflict of interest if you recommend citation of your own papers.

If your paper is objectively relevant, that trumps such concerns. Having published a related paper makes you a good reviewer of this paper. But your review would be less useful if you couldn't point out related papers only because you authored these. As long as you keep in mind that this is a slippery slope and are aware of and transparent about your conflict of interest, there shouldn't be an issue.

So, in the comments to the authors, I would explain in more detail than usual, why the paper is relevant. I would also in the comments to the editor point out that and why you recommended one of your own papers to the authors.

I review quite a lot of papers. I had such a case only once. In general, I am somewhat sceptical about those batches of "please also cite" in a reviewer's recommendation with one particular name on all of them. Some journals are also actively working against such requests from reviewers.

So, to avoid all those ethical issues and to be absolutely sure it's not my only judgment, I did this:

Mention it to the editor

In quite all review forms there are two fields. Comments to the author (the actual review) and comments to the editor that are not shown to the author. I wrote in the review something along the lines

It is not reasonable to say in line 1234 that the presented method is a "novel breakthrough", because there have been other methods better in performance (but totally different in nature). A reference to such a paper is available from the editor.

In the editor comments I wrote something like:

I would like the authors to compare their approach to the method XYZ [citation]. I find it relevant and of interest. However, as I coauthored this paper, I leave it at your discretion to communicate or not to communicate this citation to the authors.

By the way, my paper was not mentioned in the decision letter, but reviewed paper was rejected, so it did not matter much.