How do you deal with making coding errors at work?
Both to learn from the errors and minimize future ones, and also to not feel so terrible about making them.
If you are recognising the error, you now have something that you didn't know before. Also your tooling and process should be sufficient that a single error shouldn't have a negative impact on the business, it should ideally be caught in testing or code review. Sometimes it can be productive to think about how bad code can break the product and use that to feed back into your development process.
Not finding any errors would be more disconcerting, there must be some in almost every case.
The main thing is to know if your error rate and development methodology matches the risk profile, eg is it a train braking system or a playlist enhancer and did you achieve acceptable results in reasonable time.
The most important part is to be honest with them and start working on a fix as soon as possible.
Take it as a growing lesson.