Is the site still working/displaying properly , other than this error on the front end?
It looks to me like your new hosting environment has warning and notice reporting ON.
Consider looking in your wp-config.php file for WP_DEBUG and make sure it’s set to false.
Depending on how much control you have on your hosting environment, you should contact your hosting provider to adjust the PHP error_reporting to only errors.
EDIT: I would first suggest going into your development location and looking in your wp-config.php file for WP_DEBUG and make sure it’s set to true.
The best way to avoid these warnings is to fix them at the source. It’s probably a very simple fix – it looks like the function twentyten_widgets_init doesn’t exist within your theme, which if you forked the twentyten theme instead of creating a child theme, is likely the case. You may have mistakenly renamed the function without addressing the hook where it’s being attached.