p1,k2,p1,repeat around until last 8 st, which will be worked as part of next row. (The next row is a 16 stitch repeat, and so how do I add these 8 new stitches on without throwing the repeat off by ending this next row with only half the repeat done?) It’s a hat on circulars, and with many cables.

Welcome to KnittingHelp!
It depends. You are going to offset the beginning of round but if the cast on sts are a multiple of 16 or if the subsequent row takes the offset into account, the pattern will still work out. The simplest way to think of it is that you have a new beginning of round.
Can you tell us the name of the hat or give us a link to the pattern?

Men’s Ski Hat, designer Irina Dmitrieva, on Revelry.
Thanks for any help you can give me.
The row instructions state:
Row 16: p1, k2, p1, repeat around until last 8 st, which will be worked as part of the next row.

The next row is a repeat of the previous Rows 1-15.

So Row 1 is : p1, k2, p2, t3l, t3r, p2, k2, p1, repeat around
Row 2 is : p1, k2, p3, k4, p3, k2, p1, repeat around
Row 3 is : p1, t3l, p2, c4b, p2, t3r, p1, repeat around

all the subsequent rows, 4-15, are 16 stitches repeats with various instructions. Can you see my question unless she is somehow moving the cable direction over 8 stitches to make those diamond areas on the side offset??? I thought of it being a new row start at the beginning of those 8 stitches, but then half of the pattern at the end of each row will be off? I hope I am explaining it correctly.

The cast on is a multiple of 16sts so that the entire pattern will shift over. You won’t be left with half a repeat on the next row or the following rows. Think of it as shifting the beginning of round.

I love the hat. Thanks very much for asking about it.

Yes!!Yes!!Success!!Thanks so much.
I showed the hat to my son, and he looked at it last night and in a few minutes said the same thing you just explained, but he had to show it to me visually before I got it lol. He has never knitted a stitch in his life. Thanks so much again. :woot: