AE not rendering parts of the footage

Without knowing what is going on in your comp, it's hard to have any clue about your workflow, your OS, or even the version of AE you are using. The only guess I can make with the information you provided is that the problem might start with the two MP4 files you have in the timeline. MP4 is not a production format; the format usually uses IPB interframe compression, so 2/3 of the frames are calculated from the only real frames in the footage. The usual fix is to use the Adobe Media Encoder or some other software to transcode your footage into a frame-based, visually lossless production format. If that does not fix the problem, I suggest that you dive into the pre-comps and check the modified properties of all layers (press 'uu') to look for potential issues with effects you have added to the comp. 

If you are still at a loss, you can try splitting the comp where there are cuts and render smaller portions of the comp using a production format, then edit the pieces together in Premiere Pro. If that still does not work, we need to have a complete description of your workflow and see screenshots showing the modified properties of the problem layers.