How to delete logs automatically after a certain time and restart the process that fills up the log file?

With logrotate you can configure how big a log file may get or after how much time:

• the log files are rotated (log.n becoming log.n+1, and the last log file being deleted)

• the current log file is truncated without disturbing the writing process.

Take a look at man 8 logrotate.

I guess that you start the script/program with nohup like

nohup scriptname 1>logfile.log 2>& &

I would recommend instead of deleting the log file just to clear it with

echo -n >logfile.log

If you delete/move an open file it will be written until the process will close the file or the process will end.