This question illustrates a crucial difference between fundamental fields and everyday ones.
Everyday fields are made by averaging over many particles. For example, consider a sound wave in air. It's nothing more than a ripple in the 'displacement field': it says that at one point, the air is stretched, while at some other point, it gets be squeezed.
But if you zoom all the way in, to the level of individual air molecules, you'll find that nothing is getting stretched or squished at all! In fact, if you look at a single air molecule at some moment, you won't have any idea whether it's in a sound wave or not. We only see sound waves when we average over the positions over many air molecules; the individual molecules make up the field.
Fundamental fields, like the electromagnetic field, are not the same way. Such fields are not made of anything; we do not construct them out of smaller pieces! In particular, the electromagnetic field is not "made of" photons like air is made of air molecules. This misconception suggests that if you got rid of all the photons in a region, the electromagnetic field would cease to exist there, which is totally false.
A better analogy is that photons are like sound waves. You can quiet a room, but the air will still be there; similarly, the electromagnetic field as an entity still exists when there are no photons.
So, to answer your question: the electromagnetic field carries electromagnetic waves. The waves are made of photons; they are not being carried by photons.