How can I change the rate at which Gmail checks for new emails?

The new mail system does not use the classical POP/IMAP wherein there was a scheduled check on the server for new messages. Now, the notifications are directly pushed to the device by using the Google Play Services infrastructure.
Most Android phones use Doze to preserve battery life. That is if the device is not being used for some time, it will enter a battery saving mode wherein it will not actively listen to incoming push notifications, instead it will wait for some time to do a routine check. As soon as you move the phone, it should wake up and do an instant check.

Most phones will have a way to partially allow the apps to continue to work normally under the lockscreen. The settings will depend on your device, but you can try looking into: Settings > Apps & notifications > Special app access > Battery Optimization. In there, disable the battery optimization for Gmail app.

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