Cryptic crossword clue
Watches a fly going this way and that 3-6

The most common and most likely answer to this crossword clue is the 10 letter word SEE-SAWING

I've seen this crossword clue in The Guardian
S
E
E
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S
A
W
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G

Buddy explains!

The straight part
We know that "going this way and that" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case going this way and that = see-sawing.
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.

Crossword Answer definitions

SEE-SAWING verb
  1. To use a seesaw.
  2. (by extension) To fluctuate.
  3. To cause to move backward and forward in seesaw fashion.

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