Yikes!
I would look at your plant very closely under bright light and use a magnifier to see if i could find any bugs.

I would also google plant pests or possibly reddish brown deposits on tops of leaves, and see if anything looks like your problem

I would fill a spray bottle with water then add a couple of drops of Dr. brommers pepperment soap and forcefully spray the plant. In a couple of days I might add a couple of teaspoons of etoh and or peroxide to a spray bottle of water and spray the plant again. If I really thought I had bugs, I would repeat the soap spray.

I think this plant might need more light

Ralph, discarding the usual pests; scale insect, mealybugs, whiteflies and spider mites, the thing that best is described as reddish brown is 'aphids'.
What you see might be the eggs or larvae before the stage of adulthood.
The black you see might be the beginnings of sooty mold from the excretion of honeydew.
This definitely should be treated and there are a number of ways to begin.
The first thing, if you can see them, try to pick them off.
Otherwise, take the plant somewhere where you can spray indiscriminately to knock the pests off. Once off, they cant climb back.
But, like other pests, they fast bear young and so you should try to break the cycle from egg, to larvae to adult.
This is usually repeated 3 times over a 10=day period.
The other method prescribed is soap (not detergent) and water. I recommend a 40 - 1 solution.
Your imperial quart is 40 oz and if you have such spray bottle, it probably has such measure.
Forty parts water/1 part soap, spray the plant top to bottom, bottom to top ensuring all parts of the plant is wetted. Wait 10 minutes, then rinse the plant with room temp water top to bottom, bottom to top. You can treat the soil as well. To avoid over watering during such application, simply either cover it or tilt it letting water run off. The soil though should be disturbed to uncover any eggs or larvae therein.

The one thing against Anthurium is it needs constant attention and care. People compare such need to orchids.
You have it in the office. You don't say what kind of light reaches it -- from what source.
The plant, while not liking direct light, it does like good light. That speaks for natural sunlight.
Some offices are the last place a plant should be thought to be brought into.
Do you depend on fluorescent light...and how far away is it.

Anthurium does prefer high humidity....the pot of water will help. A moisture tray might be a better way.
The humidity around the plant should be in the neighborhood of 50 percent...or higher. Some people working in an office having such high humidity would probably complain about what its doing for their health.
Its nice to have plants in offices....but I think this one is better suited for a home where its care is better suited.
For instance, besides humidity around the plant, it is directly prescribed to be misted often EVERY DAY.
I take it you don't live in your office and that such care is out of the question.
For this reason alone, Anthurium is not what I'd suggest for an office--one that you spend the usual 8-hour day in.
I recommend you consider something along the lines of pothos or philadrendron that can take less care.

The office I assume too has doors...from outside and to other parts of the building. Drafts abound possibly in such office. Anthurium does not like drafts.
As well, I assume too that the office, on weekends, are not so high heated as during weekdays. This plant prefers higher than normal temperature--generally in 80 - 90º F range ...F....27º - 32ºCelsius.
It can get along with less as long as such temperature is constant, it will adapt. What might such temperature of the office be on weekends?

Its medium should be a porous one with leaf mold and coarse sand. From this you can see it drains very well.
Such water given it must be at room temperature.
During its growing season, it should be fertilized often, generally about every 4 weeks.
When its given food, the water soluble fertilizer would follow it being watered first.

Thanks to both of you, really helpful!
While I feel sometimes as though I live in my office, it's not actually the case! So the Anthurium has been transferred home and switched for my Amaryllis, which is needing a tether as its flowering very high this year!
I'm fortunate to have very good sunlight in both home and office, but as you both point out my Anthurium is going to need TLC, so I'm off to buy a couple of spray bottles!
Thank You!!