You should always test, irrispective of the job...the question is the amount and type of testing...With faulty equipment, you need to try to establish what caused the fault. Is it the installation, the equipment itself, the environment, etc. Is the equipement installed fit for purpose, i.e. will you replace it, for it to fail soon after due to what it is doing, or the external effects on the equipment etc.
Once you have established the cause of the faults, then you can establish the necessary inspection/testing required once rectified.
Out of interest, you mention that you have Part P...can you expand on this.
Yooj