A long time ago it was assumed PO4 (ie Dissolved Inorganic Phosphorus or DOP) had to kept below .03 mg/l to help control algae. Unfortunately this was not supported be research.* In reality reefs are dealing PO4 levels as high as .3 mg/l due to upwelling and they are also playing with Dissolved Organic Phosphorus (DOP) as well as Particulate Organic Phosphorus (POP). It was also assumed corals and algae compliment each other but in reality they are competing with each other for phosphate (and other nutrients) and nuisance algae often will win out, especially if there's a lack of herbivores to feed on it and put the nutients back into thr food webs in a system. (The research on Dissolved Organic Carbon, DOC, argues there's also complex chemical and biological warefare going on but htat is a far more complex subjet still being parsed be researchers.)
So, what I would do if this was my tank, get a good phosphate test kit and find out what your PO4 level is, I'm pretty sure you'll need to raise it and the best way is to feed your fish more. Research has show the MINIMUM level corals need is .03 mg/l and higher is better to assure you don't drop below that level# Get one or two of the short spine urchins (yes, some individuals will carry around frags that aren'e securely attached but urchins are essential for reefs) my prefference are Royal and Tuxedo. As mentioned above get some wild or maricultured live rock as it provides sponge and microbial stuff that's impossible to culture and stick into a bottle+. Don't do more than 5% or 10% water changes weekly and you can help the urchins by using steel straws to scrap algae off the rock when you do water changes. BE PATIENT! You are making funamental changes to the ecosystem that will benefit corals and deter algae, yu can't make it happen over night or even in a week or two, this is in reality an ongoing process as corals and algae will always be competing with each other.
* "Nutrient over-enrichment is considered a major cause of this decline because degraded coral reefs generally exhibit a shift from high coral cover (low algal cover) to low coral cover with an accompanying high cover and biomass of fleshy algae. Support for such claims is equivocal at best." Szmant 2002
+ Aquabiomics Article here on R2R
# https://www.reef2reef.com/threads/what-is-your-n-p-ratio.994106/post-11520389
Here's fig 4 from this paper looking at how the different forms of phosphorus are processed by corals. As you can see it's pretty complex and depending on the food webs in a reef system it's possible corals can actually be a source of PO4 (DOP).
