(Int = fitted model intercept)
Testing the association of NII-RF and ME/CFS without covariates:
p-value of ME/CFS derived by comparing difference in model performance between:
model1: NII-RF ~ Int
model2: NII-RF ~ Int + Beta1 * ME/CFS (0/1 binary)
model1 and model2 each have their own precision. If a model containing ME/CFS is better at predicting NII-RF scores than a model containing just the intercept, we say there is a significant association.
Testing the association of NII-RF and ME/CFS with covariates:
p-value of ME/CFS derived by comparing difference in model performance between:
model3: NII-RF ~ Int + Beta2 * age
model4: NII-RF ~ Int + Beta1 * ME/CFS (0/1 binary) + Beta2 * age
If the model performance improves when the ME/CFS variable is added, we say that ME/CFS is significantly associated with NII-RF accounting for age.
If ME/CFS has any predictive power for NII-RF, you will get p < 0.05 comparing models 1 and 2. If you want to ask whether that association is confounded by age, you compare models 3 and 4. The improvement in precision gained by adding age would be present in both models, so it doesn't matter.
[Edit: and for ME/CFS to be significant in the model3 vs. model4 comparison, it by definition has to have good predictive power for NII-RF. So you see why it would be weird to get significance comparing models 3 and 4, but no significance comparing models 1 and 2 (for all the measurements except NII-RF, which was significant in both comparisons)]