Thomas J. answered 8d
Experienced Social Studies Tutor (History & Civics)
This depends first on what you consider "the Constitution." If it's strictly Articles 1-7, then the easy answer to this question is that the biggest shortcoming was that while the Constitution prescribed what powers Government had (enumerated powers), it did not in any way restrict the federal government from specific areas or actions. The fix is what we call the Bill of Rights. This could have been incorporated into the language of the Constitution itself when written, and in effect, it was but like I said...this is the easy answer (probably too easy).
Outside of that, the obvious candidates are everything the subsequent Amendments address - with the single biggest without a shadow of a doubt being slavery and the issues surrounding it, e.g. the right of all persons to vote, be treated equally, represented/counted equally and considered full citizens.