Yes, a commercially available, third party USB Power Adaptor can be used to charge your music player.
Make sure to select a USB Power Adaptor that has an output current of maximum 1 A, and a voltage of 5 V.
There are USB cables available which are being promoted as "For Charging Only". These cables have a reduced internal connection pattern, in which the D+/- communication wires are not connected.
For Bluetooth and noise-cancelling headphones, but also for most other Sony portable products, it is important that these communication lines are connected. If they are not connected, charging your device may become impossible or problematic.
In case your cable needs replacing, make sure that you purchase an original Sony cable at a Sony Authorised Centre, or that the cable is a "fully connected" standard USB cable.
IMPORTANT: Charging via a USB AC adapter outside of these specifications is not covered by the warranty.