Sony NW-A306 aptX and reconnecting issue
It seems like switching Bluetooth connection quality requires forgetting the device. If you switch connection quality to (e.g. aptX HD) and reconnect the device it can have issues while reconnecting when powered on. In my case, I had to go to Bluetooth settings every time headphones where turned on and manually pair them.
To change sound quality select
Settings -> Connected devices -> Connection preferences -> Bluetooth -> Wireless playback quality
and select whatever preferences you like (make sure to double check whether your headphones support selected codec).

In order to forget connected device select
Settings -> Connected devices
choose settings for a given device

and make sure to forget it

Once I have forgotten the device at Sony NW-A306 and reconnected it from the scratch it works as expected. When I power on headphones they are connected with a player automatically.