Currently if there are custom fields created via a connect app, values for these connect add-on Custom field cannot be set directly via text.
We can only set values via the custom field value ID's.
It would be great if Jira API supports listing of values that can be set for such field directly or atleast be able to set text values like we set values of custom fields directly created in Jira.

Workaround
Use the Get Issue metadata API to get the values that can be set for these custom field.
Capture the ID.
Use this ID to set the custom field value via Edit Issue API via the following code block:

{
    "fields": {
        "customfield_xxxxx" : ["<id_captured from_get_edit_metadata>"]
    }
}