A very warm welcome to the HP community. A very good day to you.
I understand that you would like to install the drivers for your HP Photosmart 338 Printer on your Windows 10 computer. Not to worry, I'll do my best to help you.
I did some digging on this case and found that there are no full-featured drivers available for Windows 10. However, we could try to install the inbuilt printer drivers from Windows 10 and check if that helps.
I suggest you to perform the following steps to install the inbuilt printer drivers:
Search Windows for devices, and then click Devices and Printers in the list of results.
Click Add a printer.
Figure : Clicking Add a printer
A Choose a device or printer to add to this PC window opens.
Figure : Example of printers found in the Choose a device or printer to this PC window
If your printer is in the list, select it, click Next, and then follow the on-screen instructions to install the driver. You can now use your printer.
If your printer is not in the list, continue to the next step.
Click The printer I want isn't listed, and then select Add a local printer or network printer with manual settings.
Figure : Selecting Add a printer using local or network printer with manual settings
Select Use an existing port (USB001: Virtual printer port for USB), and then click Next.
Click Windows Update, and then wait for the print drivers to update.
Click HP, Hewlett-Packard, or Hewlett Packard in the Manufacturer pane, and then click the name of your printer in the Printers pane.
Click Next, and then follow the on-screen instructions to install the driver. You can now use your printer.
Disconnect the printer USB cable from the computer, and then reconnect the cable. You can now use your printer.
NOTE: If the drivers for your printer are absent in the list or if the printer still does not work then please choose and install the drivers for HP Deskjet 990c. The drivers for HP Deskjet 990c should work as an alternate.
I hope this information helps. Let me know. All the best! 🙂
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