Here's why this sentence works:
* "Draw a long bow" means to exaggerate or tell a tall tale.
* The politician's claim of eliminating poverty in one term is highly improbable, thus suggesting exaggeration.
Here are some other examples:
* She drew a long bow when she said she could finish the project in a day.
* The salesman drew a long bow when he claimed his product would make you ten years younger.
* The news report drew a long bow when it suggested that the government was planning a coup.