It certainly sounds like a hardware issue with the battery or the battery connector on the system board could be faulty. Please try checking with a replacement battery to rule out the battery connector issue.
If Primary (internal) Battery (601) appears in the alert message, it means the measured storage capacity of the primary (internal) battery is less than 25% of the original storage capacity. The number “601” denotes the associated error code that is recorded in the system log.
Try running a battery test on your PC from F2 (System Diagnostics) and check if the battery is functioning correctly on your PC.
Refer this article to know more information about running system diagnostics on your PC.
If the battery test fails, please contact HP support for service options.