Hi, thl04!
Something goes very, very wrong. Maybe some hardware incompatibility.
My next advise (maybe silly one) is to try the older TinyCore version, even start with TinyCore 6.4.1, not necessarily Pure64 version, you can find this iso images in Downloads past releases. If you want try the minimal Core, not TinyCore niether CorePlus. Silmply to have some running system.
Maybe some kernel modules are to be blacklisted, but this means tight examining the hardware specifications of your box. By the way, what is exact model of your HP?
Maybe some linux kernel (not TinyCore specific) boot codes will repair the boot.