You do NOT need to apply again if you have another website. You can put the same code.
From the Adsense Help Center https://www.google.com/adsense/support/bin/answer.py?answer=9913&ctx=en:search&query=another+website&topic=&type=
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How do I add more sites to my account?
There's no need to open another account or notify us when you place the AdSense code on a new site. The ad code and search code generated through the guided steps on the AdSense Setup tab of your account are valid for any page in any site on which you want to display Google ads or an AdSense for search box. Simply add the code to your page source, and we'll automatically detect the new site.
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However, it is best to create a separate CHANNEL for each site so you can track how each website performs. Read about how to set up channels from the Google Adsense help
https://www.google.com/adsense/support/bin/answer.py?answer=9868&ctx=en:search&query=channels&topic=&type=f