Would a tidally locked planet have a higher or lower amount of photosynthesis compared to a planet like Earth?

It would most likely have less. (I say most likely because until we get better data on tidally locked planets we can only make predictions on how their climate might act.) There would be less photosynthesis not just because half the planet is in darkness, but because much of the daylight side is probably uninhabitable.

Simply put, the areas receiving the maximum amount of light would become so hot that they'd become vast deserts where little or no plant life could survive. The only habitable zones would be a ring bounded on the inside by the hottest areas and bounded on the outside by the night side. Oceans could help mitigate the harsh daylight side but even with them it's unlikely that the brightest areas would develop the same level of biodiversity as the habitable ring.

As an interesting side note, any plants that did manage to survive in the bright areas would probably evolve white leaves to reflect as much light as possible to avoid being scorched.

The availability of sunlight is not the main factor limiting photosynthetic production.

The main number one limitation of photosynthetic production on Earth in the current geological age is the extremely low amount of carbon dioxide in the air. Basically, plants ate it all in the Carboniferous period which ended 300 million years ago.

The number two limitation is the high concentration of oxygen; oxygen is poisonous to the enzymes which work in the photosynthesis process.

The number three limitation is the scarcity of water in a large part of the planet's surface.

The number four limitation is the scarcity of phosphates in the soil in most parts of the world. That's why phosphate fertilizers work so well to increase agricultural production.

The availability of sunlight is only the number five limitation.

Probably less photosynthesis (but not necessarily less CO$_{2}$)

Mandatory disclaimer: We know a fair bit about exactly one life-bearing world (Earth) and that world is not tidally locked. So this answer could be significantly wrong.

A problem with a tidally locked world is that while atmospheric action will cause some variation, each area maintains basically the same temperature all of the time. Let's take Earth as an example. If it magically stopped spinning (with everything magically slowed down at the same rate to avoid Earth-scouring winds destroying everything) and became tidally locked to the sun with the Democratic Republic of Congo experiencing permanent midday, then the DRC would attain a temperature close to the boiling point of water and become uninhabitable. Temperatures in Tripoli and Capetown would probably still be above human habitability, but some plants might be able to survive. The survivable land areas for most life as we know it would be parts of Europe, Iran and Iraq. The eastern tip of Brazil, much of India and northern Europe would have a permanent climate equivalent to summer within the polar regions today. (Everyone in the opposite hemisphere - Australia, the rest of the Americas etc - has frozen to death, obviously.) Big chunks of the Atlantic Ocean will have survivable temperatures, but there is unlikely to be much photosynthesis occurring there.

Alternatively, if a spot in the middle of the southern Indian Ocean became the spot with permanent midday, then there is far more land in the mid-to-high latitudes where temperatures are tolerable. This assumes that the vaporised Indian Ocean eventually recondenses and freezes on the cold side of the planet...

A third alternative - the Earth is tidally locked and either is moved into a more distant orbit or a dimmer sun is used. In this case the "permanent midday" area could be inhabitable but it gets much colder as you move further away. Taking the first example - Africa could be entirely inhabitable but southern Europe etc are really cold and eastern Brazil/western India etc are permanently sitting at -40ish temperatures. (Note that tidal locking is pretty unlikely with distant orbits, so a dim-but-massive sun is probably a good idea, though outside the scope of this question.)

Finally, none of this necessarily implies certain O$_{2}$ or CO$_{2}$ percentages. Even if there is lots of area for plants, they do not need to be as efficient in their use of sunlight when it is always "on". On the flip-side, if there are few animals then the O₂ levels could get really high to the point where thunderstorms cause massive fires. The equilibrium has been different at various times in Earth's existence, resulting in e.g. the massive insects that could exist in the Cretaceous with higher oxygen levels.

In my estimation, a tidally-locked earth-like planet would have as much, if not more photosynthesis than Earth itself.

Why would this be, when half of the planet would be in permanent daylight, and the other half would be perpetually dark?

If we were dealing with Earth itself, with its large amount of surface water, we would expect much of that water to evaporate from the daylight side and eventually be deposited on the night-time side, where it would freeze. However, ice is not static, it flows under the impetus of gravity as glaciers, so we could expect melting and evaporation of the glaciers around the twilight edge. While much of that water would eventually end up on the night-time side again, we can expect that some of it will temporarily end up falling as rain on the sunlit side.

Given that the seas on Earth are very deep, we could expect that some of them may reach around the sunlit side, and be deep enough that they don't boil dry, but remain filled with glacial meltwater from the night side and twilight zones.

Then, we can expect that photosynthetic organisms will evolve to be able to live on the daylight side. Why would they not, when there's all that light, all the time? They would have adaptations that would allow them to live in such a hot, low rainfall, permanently sunlit area. Every possible part of the sunlit surface would likely have some photosynthetic organism on it, and seas on the sunlit side would be a soup of photosynthetic organisms

However, the night-time side would be covered in ice. We would not expect much life in that area, and the little that existed would lead long, slow, low-metabolism lives, most certainly with no significant photosynthesis. We might expect under-ice black smokers with chemosynthetic organisms clustering around them, living at a faster rate than the ice-bound organisms on the ice.

So, while only half the planet would have photosynthesizing organisms on it, they would be photosynthesizing all the time, not just half the day.