Pattern and Song mode are meant to work together.

Convert the Pattern Section to a blank Song.
Then use the Song Job: "Split Song to Pattern" to convert just the measures you want back to your Pattern Section replacing the old data with the new.

This will allow you to remove the first measure and add additional measures to the end.

Say your Section is 8 measures and you want to remove measure 1. You would create a Pattern Chain with just the one Section.... Convert it to a blank Song location - this creates an 8 measure Song. Then set the "Split Song to Pattern" setting so the measure range is set to just the measures you want: 2-8 if you wish to eliminate the first measure, or 2-xx if you wish to add empty measures on the end. Set the target back to your original Pattern and Section thus replacing the old phrases with these new ones.

Thank you, Bad Mister.

I hear you basically saying that there is no option to remove a Measure in Pattern Mode on MOXF, and it has to be done in Song Mode.

How about Cubase? Can I transfer a recorded pattern to Cubase and continue there?

I hear you basically saying that there is no option to remove a Measure in Pattern Mode on MOXF, and it has to be done in Song Mode.

Well, yes. Recordings done to Pattern mode are encased in an entity called a "Phrase" that you set the Length prior to recording. This is the information that allows it to Loop Play - the length being predetermined.

To shorten or lengthen the Phrase is basically to recreate it defining the new start and end points to establish a new Loop length. This length information is kept in the header of the Phrase.

In Song mode being linear, you can simply transfer the Phrase or entire Section to its layout, then export just the Phrase Length you require establishing new Phrase Lengths back in your Pattern.

See the following article:
One Sequencer, Two Modes, Many Related Jobs

How about Cubase? Can I transfer a recorded pattern to Cubase and continue there?

You can save the data you recorded to the MOXF Sequencer as a SMF (Standard MIDI File) which can be imported to Cubase.

Thank you for details, Bad Mister.

I somewhat understand the logic you describing, but it would be helpful if there was an option to work with measures (delete, clear, insert measures) in Pattern Mode, because in my view notes make a measure, measures make a section/pattern, and sections/patterns make a song.

I will miss the ability to work with measures in Pattern Mode a lot, because when I arrange a song I first record a section in Pattern Mode, then I reuse/repeat that section multiple times with some changes in certain measures, then I add all the sections together to make a song.

What technique should I switch to from what I described above to avoid constantly converting sections into a song and back into a section to modify measures?

I think instead of recording a full section in pattern mode, and then trying to modify measures by converting to a song and splitting back to pattern, I have to record a part of the pattern/section that will be repeating, and then keep copying that section and adding/recording measures that are different.

Thank you for the link and suggestion on SMF, Bad Mister. These will certainly help in the future.