What was published was the citare (notice of proceedings telling the respondent about the hearing). The applicant has filed proof that she published that. I'm guessing she did that because of a problem in contacting him, maybe she didn't know his whereabouts?

Here's an example in English of a divorce case in which this was done:

It then transpired that shortly after filing the amended Acknowledgement of Service, she had returned to Ukraine and issued Ukrainian divorce proceedings, stating that she was unable to provide an address for service for H (...) as she could not reliably state where he was. As a result, H was served by public notice on page 15 of a Ukrainian newspaper, in a small classified advertisement in Cyrillic.
https://www.familylawweek.co.uk/site.aspx?i=ed128253

Relevant provisions of the Romanian Code of Civil Procedure:
https://lege5.ro/gratuit/gyztaojtgy/citarea-paratului-codul-...

https://lege5.ro/gratuit/gmzdmobzgq/art-167-citarea-prin-pub...

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Note added at 33 mins (2022-05-16 11:21:06 GMT)
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No, grădiniță isn't a publication. Your sentence mentions two different and unrelated things that the applicant submitted to the court, one before the comma and one after it: (1) proof that notice of the proceedings was served on the respondent by public notice, and (2) a certificate issued by a nursery school. (1) was not published in (2), the sentence doesn't say where the notice was published.