HOME is not where the nozzle is at the center of the bed, it's where the probe is at the home position, not the nozzle. HOME is also not Z0, unless you’ve changed this, and it should be about 10mm above the bed, yet we expect it to be ~10mm +/- the gcode z offset and a variety of other factors. See Layer Height Composition for a diagram that explains some to the factors that go into the exact nozzle position.

So why's the nozzle too high or in the wrong expected position? Often it's the expectation is wrong.

The bed screws could be set at extremes. Unscrew them each, all the way, then screw them back on just 1 1/3 full rotations and relevel the bed with SCREWS_TILT_CALCULATE. Do not level with the paper method.

You could be using the gcode z offset for more than its proper role as nozzle height adjustment and also overloading it with an error adjustment for the probe not being calibrated. If your gcode z offset is negative, then this confirms your z probe isn’t calibrated. If that adjustment value was miscalculated then your position could be off.

Set your gcode z offset value to an appropriately up/down value than it is currently, using SET_GCODE_OFFSET MOVE=1 Z Z=newValue to bisect the difference until it’s about 10mm off the bed. Then calibrate your gcode z offset. Do not use the paper method. Use observation for the correct filament squish while babying stepping.

Check to make sure it’s properly aligned in the head.

Enter the follow in the Klipper console to home the probe :

G28
GET_POSITION

This should produce something similar to:

/ mcu: stepper_x:-334500 stepper_y:312527 stepper_z:-73925
// stepper: stepper_x:141.000000 stepper_y:98.000000 stepper_z:10.000250
// kinematic: X:141.000000 Y:98.000000 Z:10.000250
// toolhead: X:141.000000 Y:98.000000 Z:10.000000 E:40218.477580
// gcode: X:141.000000 Y:98.000000 Z:9.991639 E:40218.477580
// gcode base: X:0.000000 Y:0.000000 Z:10.010000 E:40219.677580
// gcode homing: X:0.000000 Y:0.000000 Z:10.010000

This tells us the toolhead (nozzle) thinks its at the X:141.000000 Y:98.000000 Z:10.000000, which is where the probe is at the center of the bed for the Neptune 4 and 4 Pro.