When should we return 4xx or 5xx status codes to the client?
4xx codes are used to tell the client that a fault has taken place on THEIR side. They should not retransmit the same request again, but fix the error first.
5xx codes tell the client something happened on the server and their request by itself was perfectly valid. The client can continue and try again with the request without modification.
If your API is trying to save a record to a database and this fails because there is an error with the database, for instance, it's not reachable, or a constraint fails, use a 5xx code (preferably  - Internal server error). Always add a response to what went wrong. This response SHOULD be displayed to the client, or if it's an automated system, it can retry again with the same request.
If you as a client receive a  status code, you can decide to try again (after a waiting-period, for a set number of times) to see if the server can handle the same request later on. Some status codes like  can return a retry-after header. This can be used to figure out how long a client SHOULD wait until a next attempt should be tried.