Problem definition
In certain situations, a module can become permanently in a pending ingress state.
Usually, the cause for this is that a requirement for the module is not fulfilled.
Link to module requirements can be found here.
Module registry example:
Module shows as pending.
When the module is selected, a screen similar to the following example is displayed.
Solution
To delete the module click on the pending button.
You will then be directed to a link similar to the following,
https://app.terraform.io/app/<Organization_Name>/registry/modules/private/<Namespace>/mod-name/<provider>/status
Select Delete module to delete the module from the Private module registry.