The most convenient way to install HotA on Linux involves Heroic Games Launcher program. It allows you to install HotA using both GOG and the standard installer (you can choose the method that is more convenient for you).
Heroic Games Launcher
Download and install Heroic Games Launcher https://heroicgameslauncher.com or from your distribution’s repository
Launch Heroic Games Launcher
Installing the GOG version
Click Login in → GOG LOGIN → log in to your GOG account
Click Library
Find and install Heroes of Might and Magic III: Horn of the Abyss
Launch the game
Installing the standard version
Click Library → ADD GAME
Click RUN INSTALLER FIRST
Select the Heroes of Might and Magic III: Complete setup file (.exe) → complete the installation
Click RUN INSTALLER FIRST again
Select the HotA setup file (.exe) → install it into the Heroes of Might and Magic III: Complete folder
Click Select Executable → select h3hota HD.exe in the HotA folder → сlick FINISH
Launch the game
Additional Notes
Use the ADD GAME button to create shortcuts for editors (e.g., the map editor).
To do this, select the editor’s .exe file in the Select Executable field.