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Pedigree
dog’s
heading
off
to
investigate
more
thoroughly
(3,6)
Pedigree dog’s heading off to investigate more thoroughly (3,6)
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The definition is at the end of the clue.
How the clue works
The definition is to investigate more thoroughly, meaning to dig deeper. The wordplay takes PEDIGREE plus D (the heading letter of "dog's") to give PEDIGREED, then "off" signals an anagram of those nine letters to produce DIG DEEPER.