You can upload Voice Memo recordings from an iPhone via one extra step:
Create and select the recording in the Voice Memo app then hit the Share icon on the left (a box with an upwards arrow)
On the options page, hit 'Add To iCloud Drive' to add it to your own Apple iCloud storage. (Any location is fine)
In LessonMate hit 'Add New' in the new lesson page and select 'iCloud Drive' from the popup menu
Locate the audio recording you just saved there and select it
That should allow you add any Voice Memo audio recordings directly to a lesson. It also works with most other files stored in iCloud Drive, and Dropbox & Google Drive if you also have those connected to your iPhone.