Is there a subtle difference here, or can we safely eliminate some words. For instance: "I stepped inside to take a peek at the work being done." VS "I stepped inside to peek at the work being done."
I think there's a slightly rougher connotation with "take a peek", maybe? Like, I'm trying to imagine a tough action hero type "peeking" at something. I can't. "Take a peek" isn't a lot more butch, to me, but I think a little.
To me taking a peek sounds like you're doing it for a much shorter period of time. When you peeking, you're standing there looking at whatever for an indeterminate period of time, whereas when you take a peek, it sounds like you're just turning the corner and/or getting however close you need to be and just check up on what the place looks like before heading back. This can be clarified by substituting the word "peek" with another, such as "look": if you take a look at something you're just doing it to check up on it or otherwise doing it only for as brief a moment as possible, while if you look at something you're doing that as long as you want to or until you're allowed to leave or whatever. So I say that if it's a routine or whatever thing that is something you can do practically instantly it should be "take a peek" and that if it's an ongoing activity where you stand somewhere and observe until you see something that is due to happen or just for whatever amount of time you have on your hands it should simply be "peek" (of course the truth in your situation is probably somewhere in between, and then you have to choose which relates the situation most accurately).
Oddly, I had the opposite reaction--"take a peek" seems cuter and more pink-ruffly than "peek." But they both seem pretty cutest; it might be better to find a completely different phrase.
My perception of peeking is that it's a bit surreptitious, like peeking at what a neighbouring commuter is playing on their tablet. I'd agree with Bjørnar Munkerud that taking a peek implies a quicker action, though peeking doesn't seem to be much longer. Neither action seems very action hero. How about "Sneaking a look"?
To take a peek may imply a quick look e.g. to see if a new window has been fitted or that no-one is slacking, whereas to peek would be to observe progress for a short while and watch a few tiles being cemented down. If the length of the peek is not crucial, leave out 'to take' in order to cut out superfluous words.