Preliminary/Warning: A correlation test is not an appropriate method for analyzing potential risk-factors!
Let's (very precisely) recall, what a risk-factor is (see Bali/Engle/Murray (2016), p.173f.), by using the size-anomaly (Banz (1981)) as an example:
The difference in expected returns between small and big stocks is due to exposures towards a latent (unobservable) priced risk factor, which are cross-sectionally highly correlated with firm size. The SMB portfolio consists of long positions in stocks with low market capitalizations and short positions in stocks with high market capitalization. This factor-mimicking portfolio is designed to generate returns that would be realized by a portfolio that is long one unit of exposure to the latent risk factor with minimal sensitivity to the other risk factors, and therefore is ideal for use in a multifactor risk model.
Although the phrase correlation is mentioned, it is not how to identify potential risk factors.
What you really need to do is similar to Fama/French (1992) and Fama/French (1993), so to build up a portfolio strategy where you sort equities based on your ESG-score. Your ESG factor-return is then the difference (i.e. hedge portfolio return) of the "high minus low" portfolio. I have described this portfolio strategy in more detail in these answers 1, 2, 3 or 4.
If your ESG-score is really priced in the cross-section of stock returns, your "high minus low" ESG-portfolio return should be both statistically and economically significant/large. However, there is a further relevant criteria which is very often not tested (although in high ranking journals...): Suppose a hypothetical risk factor ABC, which returns are $0.5 \cdot SMB + 0.5 \cdot HML$, where SMB is the size-factor and HML the value-factor. The return series of ABC definitely is highly significantly different from zero, but this factor has no new information / adds no further insight on the underlying economic sources of the return generation process. In fact, the ABC factor is spanned by SMB and HML, i.e. it is just a linear combination of both (well known) risk-factors and is therefor "explained" by their returns.
What you should apply, is a factor spanning regression, described e.g. in Fama/French (2015), Table 6:
Using four factors in regressions to explain average returns on the fifth.
Take your ESG factor-return series as the dependent variable and common risk-factor series (size, value, profitability, etc.) as independent variables. If your ESG-factor captures a new dimension of risk, it should not be explained by any other risk-factors, so the alpha of this factor-spanning regression should be statistically significant and different from zero. For more details on this method, see my extensive answer (especially section "Factor redundancy test" and Table 6) here.
Bali/Engle/Murray (2016), Empirical Asset Pricing: The Cross Section of Stock Returns, Wiley, 1.ed.
Banz (1981), The relationship between return and market value of common stocks, Journal of Financial Economics 9(1).
Fama/French (1992), The Cross‐Section of Expected Stock Returns, The Journal of Finance 27(2).
Fama/French (1993), Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33(1).
Fama/French (2015), A five-factor asset pricing model, Journal of Financial Economics, 116(1).