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"However, Mitsubishi, which co-develops the rocket, told reporters this month it hopes to launch again for the other co-developer, the Japan Aerospace Exploration Agency (JAXA), as soon as early 2024."

https://parabolicarc.com/2023/10/30/mitsubishi-announces-ambitious-launch-schedule-for-japans-unproven-h3-rocket/

The h3 launches are still just showing as planned: https://en.wikipedia.org/wiki/List_of_H-II_series_and_H3_launches#2023

Looks like they finished the investigation into the causes just this week actually. Couldn't find any mention of when they'll launch next though.
https://sorae.info/space/20230825-jaxa-h3.html

JAXA、H3ロケット試験機1号機打ち上げ失敗の原因究明に結論　再発防止対策を実施へ

JAXAは2023年8月23日、オンラインで開催された宇宙開発利用部会の調査・安全小委員会にて、「H3」ロケット試験機1号機打ち上げ失敗の原因究明結果を報告しました。

Here's a link to the actual report, but good luck with the Japanese :)
https://www.mext.go.jp/kaigisiryo/content/20230823-mxt_uchukai01-000031454_1.pdf

They say it's one of 3 possible problems: either a short circuit in the "exciter", or an overcurrent causing damage, or an overcurrent in the second stage propulsion controller. As a solution they'll just improve all the possible systems.

While electrical updates don't really require a whole new rocket to be built, it does seem like they're only gonna start just now with the fixes... I'm less confident now that they'll fly again this year. Could still happen but I don't know how fast JAXA works.

They've got 5 more on the manifest this year, so I give them decent chances in spite of the setback today! (March 7 in Japan, btw)

@Mqrius They don't say when, but it looks like they're planning to do a second flight this year, this time without payload.

https://www.jiji.com/sp/article?k=2023052400495

「Ｈ３」２号機、衛星搭載せず　リスク回避、早期打ち上げ目指す―文科省：時事ドットコム

文部科学省と宇宙航空研究開発機構（ＪＡＸＡ）は２４日、１号機の打ち上げに失敗した新型ロケット「Ｈ３」の２号機について、予定していた衛星「だいち４号」の搭載を見送る方針を固めた。２号機は固体燃料補助ロケット（ＳＲＢ）を使わない予定だったが、１号機と同じＳＲＢ付きとし、早期の打ち上げを目指すとした。同日開かれた文科省の専門家部会に報告した。