Download Scanner G4010
Tonja Witcraft
The colored grain appears to be in near-white but not pure white areas. It also looks very digital in nature, the red spots don't look like film grain. Both observations suggest it's possibly digital noise from the scanner sensor. These are negatives, so very light areas are actually from the darkest part of the negative, and in digital the dark areas is where sensors are noisiest. Maybe it's a limitation of the sensor's dynamic range. Scanning enthusiasts look for the sensors with the widest dynamic range to get the lowest noise, but dynamic range is not easy to find out from spec sheets, and the scanners with great dynamic range for film are not cheap.
The horizontal lines look like a separate issue, possibly related to the precision of the scanner stepper motor. Check the HP scan software to see if it has multiple quality modes, try the highest quality/slowest speed mode.
I have a different scanner, a pure film scanner, and it can be set to do multipass scans to average out artifacts like scan lines and sensor noise, at the cost of a much longer time per scan. If the HP scan software has a higher quality option, it may include multipass scanning, or multipass might be a separate option that you have to turn on.
As people already pointed out in the comments, VueScan is not only a good scanner tool in itself but supports a long list of old scanners. They offer a free trial so you can try it out first to see whether it works for your setup.