According to Kumārila, much can be sense-perceived. It goes without saying that sensible qualities can be sense-perceived, but Kumārila thinks that we can also sense-perceive the substance behind the sense-qualities (that is, the substrate of the sense-qualities). He also thinks that we can sense-perceive the universal inhering in the particular. Thus, when we look at a brownish cow, we are sense-perceiving its colour, the substance-cow and the universal-cow.

However, this rather generous account of perception comes with some serious and specific boundaries. Perception, to begin with, is only about the present. It cannot grasp the past nor the future. There is no yogic super-sensuous perception that would be able to grasp such features of reality.
Kumārila also denies that cognitions are self-aware (this self-awareness, or svasaṃvedana is considered to be a form of perception by Buddhist epistemologists) and rather claims that we become aware that a cognition has taken place only retrospectively, through arthāpatti. Thus, besides denying self-awareness he also denies the Naiyāyika anuvyavasāya or `apprehension of a previous cognition’, through which one becomes aware of a previous mental event. Why is anuvyavasāya not acceptable for Kumārila? Presumably because it is about something no longer present (this might be the main reason for his general denial of mānasapratyakṣa if it is about prior thoughts).
The only seeming exception is ahampratyaya `cognition of ourselves qua-I’, which grasps something other than an `external’ object. Kumārila still thinks that the I is not a construct, but something objectively real, but he claims that in that specific case we have direct access to it. How exactly is still under investigation (see my previous posts on the matter), but my current understanding is that ahampratyaya grasps the I-as-knower while it is knowing something. It cannot grasp a previous I, otherwise it would violate the boundaries of sense-perception discussed above.
Thus, as much as Kumārila is generous with regard to regular sense-perception, he is strict in denying any sort of perception beyond it.