By necessity, these mounts, which used to be more common than they are now, are less precise than a scope mounted to bases permanently affixed to the barrel and/or receiver.
The trick with these mounts is that they must have some play in the pivot, but not too much.  The lock-down mechanism should be of such design, and sufficiently tight to eliminate most, if not all scope movement, while still allowing the scope to be flipped up quickly.
Check the tightness of the whole mount with relation to the receiver.  Next, inspect the pivot for excessive radial (side-to-side) play in the pivot.  Perhaps a new shaft, with less clearance, can be installed in place of the old one. Inspect closely the lock-down mechanism, and try to adjust it to compensate for the inherent play in the pivot.  Maybe  you will have to fab a new spring or two to do this.
These mounts are still sold; check Brownell's.
Mounts like this were meant to allow quick alternation between scope and iron sights.  If your iron sights are not good, useable ones, then the idea behind the mount is pointless, and you would be better off with a standard, fixed mount.
These mounts are very handy on a flat-shooting rifle that transitions frequently from open to closed terrain and back again.  On the right gun, in the right conditions, it is a real asset.