Tried your suggestion thanks siim but it didn't solve my problem - it just turns both favourites and alerts on or off together. I did work out how to do it though:
Click on the favourite/alert symbol on the map.
The favourite name will appear in the search box
There will be a solid white heart next to the search box - click on it
Select 'remove from favourites'
The heart will turn from solid white to a white outline and your favourite (heart) icon will disappear from the map and leave the named alert (alarm bell) symbol there.