Well best thing is to give them a good inspection first. Look for tears or cracks starting to form in the rubber. These can cause random misfires too.
Most popular places are the PCV breather hoses under the intake manifold from the crankshaft breather up to the valve cover and back toward the PRV on top of the turbo inlet pipe. I would replace these with something like the 034motorsports silicon hose kit and new PCV valve and be done with that. If they have not been replaced on your 03 they will likely start to fall apart if you start tugging on them too hard. The 034motorsports kit
runs around $100 + shipping. Make sure you get the right size Y-shape breather hose that attaches to the valve cover for your year - you can measure it or look on the VW part # on the hose itself (BH or AN). Eurojet makes a set too but the 034 set is quite comprehensive.
Also, inspect directly under the metal plate in front of the intake manifold. The plate sits around the dipstick - under it is a set of vacuum hoses,suction jet pump and a couple of solenoid valves. One of the solenoid valves deals with the secondary air injection pump. Check out all those hoses - a replacement harness (OEM) runs around $155 but you can replace the hoses individually with appropriate size silicon hoses. Check the F-shaped pipe that attaches to the right side of the intake manifold too. There are some other hoses and hard lines that sit around the combi valve too.
I've never done it but you can pressurize the system to look for leaks by pressurizing through the air intake hose up to about 10 PSI I think. That way you can look for boost leaks too. There are some DIYs on the web for that.
You could fit a boost/vacuum gauge - vacuum should sit around -20 on the gauge if all is well.
The secondary air pump only comes on for a cold start and then only for a few minutes - it helps emissions by assisting the engine to burn more lean at start up I think. it runs for a few minutes but then I think it runs for a short burst that is monitored by the front 02 oxygen sensor. if it's not sensing the leaner condition it flags an SAI error. Problems there could be the fuse to the pump, air pipes broken or disconnected,the SAI pump itself, combi valve
or any combination of any or all of the above. You can check the pump is running on a cold start by disconnecting the return line to the combi valve and see if it's blowing air through.