You can manage your control groups using the mouse - to some extent. To do that, Control Groups seting in Options -> Gameplay should be set to "Normal".
- Left click on a control group tab selects units from the group (what is done by pressing a number key on the keyboard).
- Right click on a control group tab assigns currently selected units to the group (what is done by pressing Ctrl-number on the keyboard).
Those commands appear to be all one can do with the mouse, neither left nor right click are modified by Shift, Alt or Ctrl keys.
Considering the above, the actions to add units to an existing control group would be:
- recall the control group by left-clicking its tab,
- Shift-select newly drafted units,
- assign the control group by right-clicking its tab.
Simply adding units when you already have only new ones selected is impossible.