WASHINGTON - When President Clinton sat down to read "A Visit From St. Nicholas" to schoolchildren at the White House, the first thing they wanted to know was whether he had brought along his new dog, Buddy. "Is my dog here?" Clinton asked, echoing one of the 60 children from three Washington schools invited to the annual holiday reading. "My dog is here at the White House," he replied. But neither Buddy nor the Clintons' cat, Socks, turned up for the reading or the round of seasonal songs that followed. The president and his wife, Hillary Rodham Clinton, sang "Rudolph the Red-Nosed Reindeer" and "Jingle Bells" enthusiastically but chimed in more softly on "The Twelve Days of Christmas," apparently having trouble with the words.