You need to set the anchor point for each region. The anchor is what does the snapping to the bar/beat/division grid, and doesn't necessarily need to be at the start of a region.

First, turn "search zero crossings" OFF in the Arrange window's "Audio" menu, as this may interfere with the exact placement/trimming/looping of audio (you can fix any clicks at region start/end with a fade up/down of 5 or so).

Double-click a region in the Arrange window to open it in the Sample Editor. In the Sample Editor's Edit menu, make sure "Update Arrange Position" is not checked. In the wave display you'll see the start point "S" and a little black triangle, which is the anchor point. Drag the anchor to the desired peak of your reference click/drum part. Close the Sample Editor - you'll notice that the region has moved to place the anchor on the bar/beat/division grid, rather than the region start. The anchor is shown as a dotted vertical line on the region.

Repeat for all other regions with reference clicks, and drag them where you need them. They'll move in steps of whatever the beat value is if you just click & drag, steps of the division value if you hold down Ctrl [Alt] after clicking, steps of one tick if you hold down Ctrl-Shift [Alt-Shift] after clicking. To move a region by individual samples, you must move the anchor point in the Sample Editor, as the Arrange window's maximum resolution is not that fine.

Note that if a region is placed a bit off the grid and you move it by dragging, it'll move by exact bars/beats/divisions - i.e. it will still be slightly off the grid by the same amount after the move - it won't snap exactly to the grid, thus allowing the precise timing to be maintained.

To fine-tune placing, it's often useful to open the "Event Float" (Options menu), which shows numerically the exact placement of the anchor point and exact length of any selected region (bars/beats/divisions/ticks). You can double-click in the event float and type in values (if you just want "5 1 1 1" for example you only need to type "5"), or click & drag up/down on any individual sub-value.