If the puddles form during or very shortly after the heavy rain started, there may be a grading problem around your foundation, combined with hydrostatic pressure that finds weaknesses in your concrete floor, perhaps in the form of cracks.
We recommend having SafeBase™ Wall Anchors installed on your bowing walls. It makes use of proven engineering methods to stabilize and strengthen your failing basement wall(s). It doesn't matter what type of foundation you have - concrete, tile, or cinder block - this system will work with any material. The anchors can be installed quickly and they will be put deep in the earth in stable, load-bearing strata.
Another solution we recommend is having Rhino's Carbon Fiber wall supports installed on the bowed walls. Rhino's carbon fiber straps will permanently repair cracked or bowing foundation walls without steel I-beams or tie-backs, leaving your walls with no unsightly obstructions and ready to paint.
Efflorescence is a common problem in concrete and masonry block foundations. The white fuzzy stuff you see along the inside and outside of your basement wall is efflorescence. Efflorescence is simply salt and can be easily removed.
However, if you see efflorescence it means you have a moisture problem and if gone untreated can cause deterioration to the basement walls.