When you save a pose as an asset, Blender creates a single-frame action with the same name as that pose asset. All of the transformations associated with that pose asset are stored as keyframes at whatever time point the playhead was at when the pose asset was created. This is convenient since you can then use your poses in animations.
So one way you can update a pose asset is to simply update the relevant keyframes. Add a Dope Sheet Editor Type area to your current workspace and change the Editing Context to Action Editor. Then select the pose asset you want to update from the Browse Action dropdown. You should see all of the bones included in the pose asset and their keyframes at some frame in the timeline.
Make sure the playhead is at that same point in the timeline (otherwise you'll create a non-static action, not a pose). Enter Pose Mode and make any changes you want to your pose, without moving the playhead (since this will clear any changes and revert back to the previous pose asset). Then, with your cursor over the timeline in the Dope Sheet, press "i" to update the keyframes. You can choose "All Channels" to overwrite the transformations of all the bones in the pose asset or "Only Selected Channels" to only overwrite the transformations for selected bones.