I need help, no clue what we are doing.

You can purchase CYA (aka stabilizer, conditioner) powder at Home Depot or maybe Walmart. Typically a 5# jug costing under $20. You then calculate, using PoolMath, how much powder is needed to raise your CYA (assuming you are doing a complete drain) by 30 ppm in your volume of water.

Add the CYA by using what we call the sock method. Get an old sock (no holes) and put the measured amount of CYA you need. Tie the sock closed and either suspend it in front of a return (hang it from your brush pole works with something weighted on the deck) OR if it still allows flow thru the skimmer, place the sock in the skimmer. Every 15/20 minutes give the sock a squeeze to speed up the process. Please DO NOT follow the directions on the container of CYA to just pour it into the skimmer and don't just toss the sock in the pool as the CYA is acidic and can bleach/stain pool finishes.
 
To confirm, earlier this month you were treating the pool for algae by doing a SLAM Process. Did you ever fully complete that process by passing all 3 criteria on the SLAM page? That would be good to confirm first. If so, and with a CYA of 30, you should let the FC fall a bit for normal everyday use (4-6 range). Always refer to the [FC/CYA][/FC/CYA] to see what your FC should be.

Your pH would appear to be at the top-end of the range at 7.8, however keep in mind when the FC is over 10, the pH tends to increase because of the reaction with the additional chlorine. Once the FC drops below 10, re-test the pH to confirm.

As for the spots themselves, it would be best to see a full set of TF-100 results, not just the CYA, FC and pH. As others indicated above, it could be the age/health of your plaster. Sometimes spotting could be residual metals in the water precipitating when the chlorine was elevated, but the pool store test showed no metals in the water. They've been wrong before, so time will tell. Please post your results as follows:

FC
CC
CYA
PH
TA
CH

This will help us narrow down any potential causes and give us a good way forward to assist.
 
Thank you, ok here are all the TF 100 results and yes it passed all 3 criteria of a successful SLAM.

FC- 16
FC- 16
CC- 0
CYA 40
PH 7.8
TA 110
CH 150 ppm

What do you recommend? We are thinking maybe the pool needs to be re plastered but hoping we can delay that a year at least. When my husband pressure washed it, a little piece chipped off and we saw the black algae was deep inside underneath the plaster.
 
It sounds as if the plaster is getting weak in some areas and is a bit compromised. Really not much you can do for that other than to maintain good water chemistry and try to prolong things until your budget allows a re-plaster job. As for the water results, they look very good. You are actually still at a SLAM level, so no need to add any bleach for a couple days and just let the FC fall down to the 4-6 range.

Focusing just on plaster, your water is in an ideal position (balanced - slightly negative "CSI"). If you don't know much about CSI, just look at that row on the PoolMath calculator. While most people (especially in AZ) are concerned about hard water and scale, they avoid having the CSI creep upwards. In your case, you don't want it going to low or becoming corrosive which over time could eat-away plaster. Right now it is not. Even though your CH is a bit low, the elevated TA and pH compensate to keep the CSI healthy. As time goes on, I expect water top-offs of hard water will slowly cause the CH to climb as well. I don't want you to become obsessed with CSI, just become familiar with it on the PoolMath so you understand how pH, TA, and CH (along with water temp) effect it. For now, you are doing well.

Don't get too aggressive with brushing on any weak plaster areas, and keep the FC at a good level to help minimize algae growth in any open areas of the plaster.
 
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