Do I leave the meat thermometer in the meat the entire cooking time?
My experience with metal probes left in meat for the cooking period, if it is not a short cooking time, can give higher than accurate readings unless the item is large enough to bury a high percentage of the probe and still hit that optimal middle of thickest area without hitting bone.  If too shallow, the metal can transfer heat itself to the measuring point so give a higher than true internal temp.  This tends to happen more often for me with less dense meat. When this is a concern, I would suggest using the probe initially to get to temp, then moving it to a new location when it reads that you have reached temperature.  It this thermal transfer via the metal is what is occurring for you, then the new positions should quickly drop to the true temperature and you will know you need addition time.  The 2nd point is usually used for less time, so does not tend to have the same false rise.

All of this is predicated on using a calibrated device of a type of probe designed to allow leaving in as mentioned in answers and comments, i.e. not plastic, not instant read, etc.  Those designed to remain in are often the ones with flexible cords to an out of oven base for reading and some wireless models.  There are those that the readout is not remote, but those have the disadvantage of needing to open the oven.