If Woodpeckers are actively seeking bugs on your tree, it may have an infestation of some kind. Woodpeckers can hear insects under the bark and in dead wood, and that is what attracts them.
I copied this from the ArboristSite.com:
"Woodpeckers feed on carpenter worm larvae. Their presence may be an indicator of borer activity. This borer tunnels into the center of the tree trunk or a large limb. This is the escape tunnel. The feeding tunnel is in the heartwood close to the bark or cambium layer. Woodpeckers trying to catch a borer larva sometimes make a line of several holes in the bark over the feeding tunnel.
Carpenter worm infestations are often overlooked. They can be identified by falling major limbs with dark tunnels visible at the break, sawdust at the base of the tree, and an opening plugged with wet sawdust Fungus feeding fruit flies, and dried fruit beetles were seen feeding in large numbers on carpenter worm openings. These insects carry plant pathogenic fungi. Dead major limbs could identify borer infestations."
I'm not an arborist, but I am a maple syrup producer in NY. When the bark is coming off of a tree, and there are woodpecker holes, along with dying limbs, it usually means the tree is dying. Call your local Cooperative Extension office (your county office will have their number), and ask if they can help you identify what bug(s) are attacking your tree, and if the tree needs to come down for safety reasons.