StrikerÂ Nicke Kabamba scored twice but it wasn't enough to halt the unbeaten Vanarama National League leaders.
Two goalsÂ in five minutesÂ at the start of the second half proved decisive.
Due to injuries Bradbury was forced to field a patched-up side minus four key players.
Brian Stock, Wes Fogden, Jordan Rose and Mike Carter were all missing from the line-up.
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Despite this the visitors more than matched their hosts for long periods of the contest.
Bradbury felt once again that his team missed out on any good fortune that was going.
He was adamant they deserved to get something from the game.
The Hawks boss said: '˜The boys gave everything. We came here and gave the in-form team in the league a real good game.
'˜I don't think they will have many harder games. It would have been a big ask to come here with with a full-strength side.
'˜We had to do it with four important players missing. All the lads who came in did well and there were some big performances out there.
'˜We set up well, the formationÂ played worked and we gave them lots of problems.
'˜In the end we should have got something out of the game.'
Kabamba gave the Hawks a terrific start putting them in front after six minutes with a good strike.
Harrogate hit back before the break after being awarded a contentious corner.
When the Hawks failed to clear Callum Howe headed home unchallenged.
Two quick goals at the start of the second half put the hosts in the ascendancy.
But the Hawks showed great spirit to get themselves back into the game on 63 minutes with a second Kabamba goal.
Bradbury added: '˜We gave it a real go and spent the last 15 minutes camped in their box.
Two-goal Kabamba was once again the Hawks star of the show.
Apart from his goals the striker led the line well giving further evidence of his increasing confidence.
Bradbury is impressed with the progress he is showing.
'˜The boss added: '˜Nicke was fantastic for us again. The goals were a just reward for all the hard work he put in.
'˜For his second he showed great composure to dink it over the goalkeeper.
'˜That is four goals he has scored and he is in a good place.'
The defeat leaves the Hawks in 20th position one place above the dropÂ zone.