Sometimes you will notice when adding gift options from within the cart drawer, the page will automatically refresh after the customer selects or changes their gifting options.
This is usually required if the cart drawer is unable to automatically refresh itself after gift options are added or removed.
The good news is there is a simple gift block setting to help with this:
Reload the page after customers save
To change it, open your gift option in the Wrapped app and press Open editor on the Gifting editor card. Under Gift blocks, press Cart drawer, then turn Reload the page after customers save on or off under Settings for this gift block. The Cart page block has the same setting.
The Reload the page after customers save switch in the Cart drawer block panel
After changing this setting, press Save, and your changes will be queued and published to your storefront.