LITTLE ROCK, Ark.—A former top aide to Bill Clinton, charged with trying to smuggle contraband to a death-row inmate, will ask a state court to allow a videotaped deposition of the man before he is executed. The request, to be made Wednesday in Pine Bluff, is the latest twist in the case of Betsey Wright, an outspoken death-penalty opponent accused of trying to take tattoo needles tucked inside a bag of Doritos chips, as well as a pocketknife and a box cutter, into a high-security prison. The videotaped deposition is necessary because her trial is set for May 25, several...