You can try below troubleshooting steps:

Check File Permissions: Ensure that the error_log file has the correct permissions for the busybox container to read it. You can check the permissions using kubectl exec to run ls -l /app/logs/error_log inside the nginx container and adjust them if needed.

Ensure that nginx flushes its logs more frequently by adjusting its configuration or by forcing log rotation before the CronJob runs and also have a look at this Techleader. pro post by John Collins to set up a simple Cron job to rotate and compress nginx log files automatically each day.

When nginx writes to a log file, it holds file descriptors open, which could cause issues when another process, like your CronJob, tries to read the file. Instead of directly reading the log file with cat, you can try using tail -f to see the log file updates like tail -f /app/logs/error_log. Also refer to this Cyberithub article on How to Check Logs of Failed CronJobs in Kubernetes Cluster.