We consider a class of stochastic heat equations driven by truncated $α$-stable white noises for $1<α<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We prove the existence of weak solution, taking values in two different spaces, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. For $1<α<2$ the weak solution is a measure-valued càdlàg process. However, for $1<α<5/3$ the weak solution is a càdlàg process taking function values, and in this case we further show that for $0<p<5/3$ the uniform $p$-th moment for $L^p$-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in $L^p$ sense and satisfies a flow property.