Because of more urgent issues which were more higher priority to handle, I spent most of the/my time with others.

"More ____" and "higher" are both comparatives, but you should use only one. Generally, short adjectives such as "high" follow the pattern "____er", and long adjectives such as "intelligent" follow the pattern "more ____" (although there is disagreement on some, such as "clever").

"Time" is missing a determiner. "Most of the time", "most of my day", "much of that evening", "some of this money", "all of those breadsticks", "none of these people", etc.

After completing the urgent works, it was almost dinner time.

"Work" is usually uncountable in English. A good countable option would be "urgent tasks".

I promised with my son to be at(that) I would be home before dinner.

This one is a bit tricky. You can say "I promised to be home before dinner", but if you use an object with "promise" (such as "my son"), you have to use a "that" clause instead.

The rest of the works made me feel sad, but I had no choice.

see above

I must say sorry to my son once I am back home/after I will beget back home.

This is another funny rule in English: even though we're talking about the future here, we *never* use "will" with time conjunctions like "before", "when", "once", or "after". Also, we don't like using "after" with "to be". "After" is better for actions ("doing"); "once" is better for states ("being").