Perhaps a bit more info in terms of which map you're looking at (there are a few options in the drop down list) and a location would be helpful.

However Googling EH & Bike route throws up the East Hampshire circular route. Is that of any help? 

Yes, it’s the East Hampshire route. The little red/blue boxes go with the red/blue dots to indicate a signposted cycle route. The box contains the abbreviation for that route. If you plan a route along it, the full route name will be shown in the turn-by-turn directions.