How to get wood veneer "pop-ups" to lay flat again on my table top?

We bought a used clawfoot dining table that we thought was solid oak. After carefully sanding it and priming with Kilz,we left it to dry overnight. Excited to paint it the next day, I first noticed several small bumps on the table top only, but thankfully not on the beautiful pedestal it sits on. Is it possible to use now, as I can't imagine setting a full glass of anything down on it,and seeing it tip over. Also do not want to see a less than level table in my dining room! Is it worth the time and cost to fix it,or do I have to write it off as a mistake, and a waste of money?

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