If you are staying in Japan with a dependent visa, you are basically regarded as a “dependent.” So, if you have a sizable wage from a part-time job, you may not be eligible for the renewal of the visa.
To cite an extreme case, say a husband is earning 6 million JPY a year with an Engineer/Specialist in Humanities/International Services Visa, whereas his wife is working for 20 hours a week under a dependent visa. If the wife is an excellent engineer and earns 10,000 JPY per hour, she ends up earning 9.6 million JPY in a year (20 hours a week x 4 weeks a month x 12 months).
In this case, the husband’s annual income is 6 million JPY and his wife’s is 9.6 million JPY. She is earning more than her husband, and thus she may be regarded as non-eligible for the status of a “dependent.”
Along with the restriction of work hours to 28 hours, you should keep your wage within the range of a dependent status so that the renewal of your visa can go smoothly.