I think the main thing is a comfort with working multi-step equations. (Doing same thing to both sides, showing your work, having sufficient care/practice to make a very low level of mistakes.) At the simplest level, think of the steps in solving for x in first order, one unknown problems. But this same trait, in other problems. I would say that this can still be drilled, emphasized "as you go" in teaching calculus. For example at a help center. And if you are teaching weak students, SHOW EVERY STEP. Get them to show every step. No "throwing something from one side to the other and flipping sign" but "subtract same from each side, showing it pedantically step by step".
Obviously there are some specific items of knowledge (e.g. exponent rules) that are important also. It's basically "most of it" from the precursor classes, so it's almost as helpful to specify what parts of the curriculum are NOT important. And this is not to say they shouldn't be taught. Just for a kid, trying to progress, with some gaps, they are low bang for the buck.
On the list of things that are lower priority (for weak kids, because life is prioritization, because they ain't ever getting to Rudin, but we'd like them to get their nursing degrees):
Most of geometry (side angle side, proofs, etc.). Only mensuration area formulas are needed.
Conic sections, except for the very basics of a centered circle and parabola and hyperbola on y-x plot. But not foci and directrix and ellipses and vertical/sideways displacements.
Most of the trig identities (just sinsq plus cossq). [If double angle or the like needed later, review it then.]
Determinants. [Review it when needed in a calc class for weak students or even just eschew entirely...it's usually "enrichment" anyhow, even within calculus or diffy screws.]
Rotation and displacement of coordinates (analytic geometry).
harder (often, but not always asterisked, sections of Alg 2): synthetic division, partial fractions, induction proofs, series, etc. For series or partial fractions, if that is part of the calculus course, make sure to cover the required algebra WHEN they do the calculus topic (and show all the steps). But don't expect precursor competence. Look at it more as an opportunity to brush up that topic they didn't handle/remember in 11th grade.
If you want a build up (versus tear down) approach, the list from Kahn Academy is decent:
Please note, I'm assuming a context of community college or other remedial work. Not precalc at Thomas Jefferson. They should have a tough course. But then they don't match Brian Rushton's posited set of students struggling with standard calc 101 because of poor algebra.
Also note: don't underestimate the importance of algebra. It's at the core of calc and diffyQs in terms of the actual working the problems. And is required in most chem, physics, engineering classes. Just to work problems (often not needing calculus even!)
Not also that this means when you DO teach a calc class for weaker students, you should be MUCH more receptive to the "here's an example, here's a method, practice small variations of the algorith, test recognizable problems versus the examples". After all, even if it is "mechanical", it gives them a golden chance to practice multistep algebra, itself.