I have spent quite a bit of time thinking about this and I have one note (maybe a very tiny partial fraction of a solution) and one question.
First, some assumptions I am making:
1. There should be one letter in each hexagon
2. The 11-letter solution will be the letters in the column with the black dots in
3. The colours of the rows don't have any specific meaning other than making it easier to see which rows the arrows connect
4. Each row will be a valid word
5. The arrows mean that the "source" row can be inserted into the "destination" row at the place indicated and make another valid word, for example the 7-letter light green row can be inserted at the start of the 4-letter light green row to make a valid 11-letter word
The note that might be a tiny part of a solution:
The only word I can think of for the bottom row, that is a valid 6-letter word that is also a valid 7-letter word when an X is added to the end of it, is SIMPLE / SIMPLEX.
This would mean that the 11-letter solution ends in L. There is such a word which relates to hexagons, namely APICULTURAL, but I have no other evidence for this being the answer.
And the question:
I'm bothered about the fact that the arrows on the hexagons come from / go to different aspects of the shape (the two left hand arrows and the one at the bottom pointing to X go from and to vertical edges of the hexagons, but this is not true for the other three on the right). Is this significant?
Based on the comments,
I notice that the first few rows at least could be months:
(lime) August / October
(yellow) August / October
(tan) August / October
Doesn't work so well for the later ones, could those be days / dates / something else?
If I'm right about the first three rows that would be an 11 letter word starting