We address a problem of sampling and reconstructing periodic piecewise polynomials based on the theory for signals with a finite rate of innovation (FRI signals) from samples acquired by a sinc kernel. This problem was discussed in a previous paper. There was, however, an error in a condition about the sinc kernel. Further, even though the signal is represented by parameters, these explicit values are not obtained. Hence, in this paper, we provide a correct condition for the sinc kernel and show the procedure. The point is that, though a periodic piecewise polynomial of degree R is defined as a signal mapped to a periodic stream of differentiated Diracs by R+1 time differentiation, the mapping is not one-to-one. Therefore, to recover the stream is not sufficient to reconstruct the original signal. To solve this problem, we use the average of the target signal, which is available because of the sinc sampling. Simulation results show the correctness of our reconstruction procedure. We also show a sampling theorem for FRI signals with derivatives of a generic known function.