There are two common pecan pests that fit the description -- the pecan weevil and the pecan shuckworm (also called hickory shuckworm). Your pecans may be infested by one or the other, or both. In any case, you can reduce the extent of the problem if you pick up and destroy fallen nuts at the end of the season, since that's where the shuckworm overwinters. It also helps to cultivate the soil under the tree to a depth of about 3 inches during late February or early March. Weevils are more difficult to control. If it's practical, you can shake the adults from the tree. Place tarps under the branches and shake them to dislodge the feeding, breeding weevils, and destroy them. These pests overwinter in the soil at a depth of up to 12 inches, making cultivation impractical and likely to damage tree roots. Fortunately, weevil infestations tend to come in cycles, becoming a problem every two or three years.