Complex sentence: Their strategy was to charge that Republican votes were fraudulently bought and to use a provision of the Missouri Constitution that provided the Speaker of the House "would count--tabulate--the votes and proclaim to the general public who won".
Split into simpler sentences: Their strategy was to charge that Republican votes were fraudulently bought. They sought to use a provision of the Missouri Constitution that allowed the Speaker of the House to "count--tabulate--the votes and proclaim to the general public who won".