While I'm sure they have extra footage in the form of alternate takes, I highly doubt more than a handful of episodes had entire scenes cut from them, let alone multiples.
No scenes do get cut short. Here's a made up example:
Picard: Set a course for the Neutral Zone. Course 321 Mark 159. Warp 9. Engage.
Helm: Aye, Sir. Course Set. Engaging at Warp 9.
Riker: Red Alert! All hands to Battle Stations. Raise Shields. Ready Phaser. Load torpedo bays.
Worf: Shield up. Phasers and Photons ready.
Picard: Set a course for the Neutral Zone. Warp 9. Engage.
Ricker: Red Alert!
Worf: Shields up. Phasers and Photons ready.
See how the second versions tightens the scene?
Many scenes have a lot more dialogue than what's shown, editing out bits of it tightens the pace, while others keep in more of the shot scene to slow things down.
While yes there are alternate shots of the same scene, bits and pieces are cut or left depending on the directors vision of the episode and/or time constraints.
In The Measure of a Man
the scenes added back did add some tension between Riker and Picard and showed more dimensions to Phillipa. At least I don't remember her acting like such a minx with Picard.