My iPhone 13 randomly has a black screen with a loading gear then takes me to the Lock Screen

Three basic troubleshooting steps to perform (test after each attempt):

A) Try a "forced" restart of your device using these instructions:--> Force restart iPhone - Apple Support

B) If you can, check your battery status in Settings>Battery>Battery Health>Maximum capacity. If it is close to 82-80%, have it replaced.

C) If the issue persists, something could have gone wrong during the latest update procedure.

I would try to install a fresh copy of iOS/iPadOS (the latest available/supported by your device model). To do this, create a full backup of your iPhone/iPad (1), then restore it to factory settings (you need a computer to perform this step) and, in the end, set it up as new (2).

When done, restore data from your previously made backup (3).

Follow the instructions provided in these support articles:

(1) How to back up your iPhone, iPad, and iPod touch – Apple Support

(2) Restore your iPhone, iPad, or iPod to factory settings – Apple Support

(3) Restore your iPhone, iPad, or iPod touch from a backup

If the issue still persists, it could be a hardware failure but we, just users like you, cannot diagnose that issue remotely, sorry.

I suggest that you make an appointment (*) and have your device inspected at an Apple Store or AASP (Apple Authorized Service Provider) among this list:--> Apple – Location Selector. After checking it out, they'll give you all the available options.

Regards

Giulio

(*) You can do it:

• Online:--> Genius Bar Reservation and Apple Support Options - Apple;
• Contacting a local Apple Store or AASP among the list provided above;
• Using the "Apple Store" app, available and downloadable from AppStore.