Using a third party app, one can quick access Telegram's Cached Files' folders. One can copy files and of course delete them.

How-to

  1. Download and install File Manager. (Link)
  2. Swipe right and select 'Internal Memory'
  3. Open the 'Telegram' folder
  4. Tap on "…" and choose 'Add to home screen'

It creates a shortcut to the path <internal memory>/Telegram and you are able to open it with one tap from your Android home screen.

Screenshots

filemanager_sidemenu.jpg

Swipe right and select 'Internal Memory'


filemanager_folder.jpg

Search for the Telegram folder and open it


filemanager_addhomescreen.jpg

Select 'Add to home screen'


filemanager_onhomescreen.jpg

… and on your homescreen a shortcut will be added.