Any DAW with MIDI capabilities should be able to handle that, and some commercial DAWs have free lite or older versions available-- e.g., PreSonus Studio One comes in a free lite edition called Studio One Prime, and Tracktion has a free older version called T5.
However, if you aren't used to working with MIDI on a piano roll editor and would find it easier to work with a musical score, you might want to consider a notation program instead of a DAW. (Many DAWs include a notation view in addition to their piano roll editor, but a notation program will generally have more sophisticated functions for notation.) As with DAWs, some commercial notation programs have free versions available-- Finale Notepad, for instance, or Forte Free. The free versions often have a lot of restrictions as far as things like how many staves (different instrument parts or MIDI channels) you can work with at once; but if your MIDI recordings basically contain just a single channel for piano, those types of restrictions might not impact you. There's also a completely free non-commercial program called MuseScore, which is loaded with all sorts of functions and might have a definite learning curve.
Either way-- piano roll editor or staff notation-- all you have to do to fix an incorrect note is click on it and drag it up or down as needed. If you're using a notation program, you might also need to add an accidental, but on a piano roll there are rows for the black keys as well as the white keys.