Repeating images in training dataset for tensorflow object detection models
Should I use the same image for multiple records?

No, because anything in the image that is not annotated as an object is classified as background, which is an implicit object type/class. So when you train your model with an image that has an object, but that object is not annotated correctly, the performance of the model decreases (because the model considers that object and other similar entities as background)

Could that be problematic when training?

Yes, this issue is going to affect the performance of the model in a bad way. In fact, a good thing to do is to add some images that do not have any objects in them and let the model be trained on them as background with no instance of a bounding box.

Would it be better if I could split said images so that they only contained one object?

Yes, this can help. Also, you can consider adding multiple bounding boxes for each image. But never leave any object without an annotated bounding box, even if the object is truncated or occluded.
I'm training a tensorflow object detection model which has been pre-trained using COCO to recognize a single type/class of objects. Some images in my dataset have multiple instances of such objects in them. 
Given that every record used in training has a single bounding box, I wonder what is the best approach to deal with the fact that my images may have more than one object of the same class in them. 

Should I use the same image for multiple records?
Could that be problematic when training?
Would it be better if I could split said images so that they only contained one object?