I would group them as follows: "Leave it to me" and "I'll take care of it" are similar general statements saying that you will take responsibility. This could be for something straightforward or something more complicated. The other four expressions imply that there is something broken that might be hard to fix. Only the 4th of your samples seems to vaguely belong in this category. In general, if the fix is something extremely simple and obvious, there wouldn't be a need to use these kinds of shorthand. "I'll make a call, clean up my desk, wipe up a spill etc." So it would sound a bit inflated to use those kinds of expressions. But if there was a problem that had a complicated fix (call multiple clients, explain a problem to each, renegotiate delivery terms etc), these expressions would sound more natural.

As far as I can see first question's answer is"l will sort it out " and second it seems to me "look at desk....." This is a little bit just like pass the ocean but choke in the lake.Just a kidding )

Your range of responses is good. However there are some subtle differences in meaning between these phrases (see below). 1. Leave it to me - (I'll be the one to fix it/take some action. Focus on "you" as the person) 2. I'll fix it - (I'll take action to remedy the problem, and usually I'd have a specific 'fix' in mind) 3. I'll figure it out - (focus is on the mental activity of thinking of a solution to the problem) 4. I'll deal with it - (same as (1), (5) and (6)) 5. I'll sort it out - (I'll come up with some kind of solution, as yet unspecified) 6. I'll take care of it - (same as (1), (4) and (5)) So for your sample problems, use the following: 1) I've forgotten to call them back (I haven't told them something important). "I'll fix it" or "I'll sort it out" 2) Look at our desk, it's a mess! (-Don't worry, I... ? ) "I'll take care of it" 3) Oh, I just spilled some water on it. Just say "I'll wipe it up" rather than a general phrase. 4) I failed to book seats for us. "I'll sort it out" or "I'll fix this".