Your child would claim this on his or her own tax return, if your child decides to file. Since your child earned less than $6300 for the year he or she would not be required to file a tax return if the income is earned with an employer and reported on a Form W-2, and there is no additional income for your child to report. However, if your child had any Federal or State taxes taken out of their check, then your child would want to file to get all of those taxes back. Please see this answer by SuperUser TomD8: https://ttlc.intuit.com/replies/5668731
Your return will not change. You will still claim your child as a dependent as you undoubtedly always have (unless by earning $5800 your 15-year old provided at least half of his or her own support, which is not likely).
For your child to file, it will require a separate return, but he or she may use TurboTax Absolute Zero, which will cost nothing as long as no extra services (like Plus or Max) are purchased. In the Personal Info tab, your child will also state that they can or will be claimed as a dependent on another return.
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Yes
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