A one-tailed test at the $5\%$ level uses $H_0: p = 0.4$ against $H_1: p > 0.4$, with $4$ passes out of $9$ observed and $P(X \ge 4) = 0.517$ under $H_0$. What is the correct conclusion?

A result is significant when its probability under $H_0$ falls below the significance level. Here $0.517$ is far above $0.05$, so a result at least this good would happen more than half the time even if nothing had changed. $H_0$ is retained: there is no evidence of an improvement. The observed proportion being slightly above $0.4$ is not by itself evidence, which is exactly what the test measures. The binomial model is valid for $n = 9$; small $n$ simply makes the test less powerful.