@oristarA,
Throwing deep or going for the long ball, is giving it all you've got. He advised Romney to give people the meat of his campaign. Show them what he really believes in. Prove that Mitt Romney is a man of the people with solid ideas how can understand the needs of the voters. Don't dance around and try to get Obama to slip up. Throw deep, show us what you're made of.
Light a cig(arette) is like saying chill. People often reach for a smoke when they're nervous or in times of stress. A cigarette calms you're nerves, or so it seems. The author is telling Obama to stand his ground and make Romney backpedal, don't lose his cool.