What's the error-correcting capacity (randomly errors, or burst errors) of polar code? Polar codes, invented by Arikan, with low encoding and decoding complexity. But the minimum distance of polar codes is not great. The normalized minimum distance goes to 0 with the block size. Yet, the existing decoding algorithms for polar codes can asymptotically achieve channel capacity.

Generally, a code with minimum distance $d$ can correct $\frac{d-1}{2}$ errors (random errors). And Polar code is also error-correcting code, I was wondering how many errors can it correct( e.g. A $[2048, 614]$ with $BEC(0.19)$ polar code.)? Much appreciated.
I am not an expert but I think you may be asking the wrong question. Polar codes don't have a meaningful "error correction capability" obtainable as in algebraic codes by (d-1)/2. Alternatively, this is not the meaningful parameter of significance.

For finite lengths their performance is usually analyzed in BEC (Binary Erasure Channel) and AWGN channels (Additive White Gaussian Noise). The well-cited paper On Finite-Length Performance of Polar Codes: Stopping Sets, Error Floor, and Concatenated Design by A. Eslami and H. Pishro-Niki on arxiv here as well as some more recent papers confirm this.