I'm probably a candidate for Brain Farts Anonymous, but I've tried various ways to move notes within a clip (it's a MIDI clip, no audio), without the clip itself being moved.  Slide just slides the clip, Nudge does the same.  It's a 4-measure clip with notes that begin about 910 ticks into the clip.  I want those notes to begin right at 1:01:000.

Piano roll view... Drag and drop.

Make sure neither the clip nor track are locked or frozen.

Both the PRV and inline PRV allow selecting notes instead of clips.

Sometimes when using the PRV, I forget I've still got the whole clip selected and hence attempting to move a note moves all the selected notes.
 
Perhaps this is what is happening to the OP ?
 
I just click on a blank area on the track area to de-select the clip/notes and then continue with note editing.

As said, midi editing should all be done in the PVR where it's very easy to move and group of notes around. If the timing is off I would simply quantize it to the best value,, like 16th. 

Thanks everyone. Drag & Drop in PRV was what I was looking for.  Surprisingly I've never needed to move notes like that before, and the Slide or Nudge function seemed most logical (select the notes in a clip and move them, right?  Nope!).  PRV is the place for that I guess.