首先,可以把这个句子翻译过来：你可以用一种轻松而自然的方式来发表评论包括一些随意而显然即兴发表的讲话.[以上为直译,意译应该是,你可以用一种轻松而自然的方式发表评论,在评论中穿插（include）一些随意而即兴的讲话.]
主干是：你可以发表评论.那么整个句子,you是主语,can deliver是主干的谓语,which是主干的宾语从句的连词,in a relaxed and unforced manner是状语,修饰怎么deliver.
这样,就可以理解前面的“Include a few casual and apparently off-the-cut remarks which”是整个宾语从句和连词的部分.从句这部分,作为一个从句,也应该是完整的.其实,它应该是“remarks include a few casul and arrarently off-the-cut”主语是remarks,include是谓语动词,后面的是宾语.所以,其实不是include remarks,而是remarks include off-the-cut.
句子也可以改为:You can deliver remarks including a few casual and apparently off-the-cut in a relaxed and unforced manner.
所以从这个句子来看,which和that都是可以的.
相当于You can deliver remarks which（that） include a few casual and apparently off-the-cut in a relaxed and unforced manner.