The problem with the instructions you have scanned is that C-C multiple bonds don’t take a suffix. They take an infix. Therefore, they don’t belong in the priority list because you always use their infix, no matter what other functional groups are present.
For example, hex-2-ene and hex-2-en-1-ol; hex-3-yne and hex-3-ynoic acid.
With respect to your name, “hexane-6-yne-2,4-diene,” you are wrong in a number of ways. Here’s the structure (# used to represent a triple bond):
CH#C-CH=CH-CH=CH2
If you have C-C multiple bonds, you never use the “an” infix. You wouldn’t say “propanene,” would you? It’s “propene.”
“en” comes before “yn,” and takes the lower number.
Multiple bonds are numbered by the lowest-number carbon participating in the bond: 1=2–3 is named 1-propene, not 2-propene.
The carbon numbering is from right to left, and this is the structure re-drawn with #1 on the left instead of the right:
CH2=CH-CH=CH-C#CH
There are six carbons, so “hex” is correct. You have double bonds beginning at the #1 and #3 carbons, and a triple bond beginning at the #5 carbon.
So the correct name is hexa-1,3-dien-5-yne. The “a” is used for ease of pronunciation; “hexdiene” is harder to pronounce than “hexadiene.” Without the “di” it would be “hexene,” without the “a.”
As an examiner, I would not take points off for “hex-1,3-dien-5-yne.”