Removing the HP Sprocket entry from the Bluetooth paired devices and adding a new printer using HP Sprocket app may resolve the issue. Let's try these steps:
Please ensure that iOS is up to date with its software, if you are unsure, you may use the steps available here to update iOS.
1.) On your iPhone, open Settings.
2.) Touch Bluetooth, below My devices, tap on the little “i” icon on the HP Sprocket device..
3.) Select Forget this Device, confirm by touching Forget Device.
4.) Turn off iPhone
5.) Reset HP Sprocket, refer to step 4 from this document.
6.) Turn on HP iPhone.
7.) Open the HP sprocket app.
8.) Tap the Menu icon , tap sprocket, and then tap Manage Printers.
9.) Tap Add New Printer, and then tap the name of your Sprocket
If issue persists, try upgrading sprocket firmware, refer to step 5 from this document.
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The above countermeasures are tried, still unable to connect. You always want to upgrade fw. HP sprocket and the iPhone without connection can upgrade? If you don't connection can also upgrade with sprocket, please tell me the way.