The most common and most likely answer to this crossword clue is the 7 letter word CINDERS
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More clues leading to the result CINDERS
Buddy explains!

The straight part
We know that "one acting dogged by drudgery" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one acting dogged by drudgery = cinders.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CINDERS mean?
CINDERS noun- Partially or mostly burnt material that results from incomplete combustion of coal or wood etc.
- An ember.
- Slag from a metal furnace.
- Any strong stimulant added to tea, soda water, etc.