I found some threads that cover this error:
» System Error - ONCoreFoundation7.dll is missing
» Photoshop CC2015.5 wont start - OnCorefoundation9.dll error msg
Apparently, the error is related to software from OnOne Software. Reinstalling the software in question should fix this.
Also see this help article from the developer: Ononecorefoundation.dll Is Missing From Your Computer- Photoshop Error Message – ON1 Support
That fixed it. Thanks very much for your help. It's very much appreciated.