Cobblers!Sounds like a worn chain and that WILL affect shifting. Check it for length.
Bike is booked in for a inner and outer cable change, fingers crossed!
Most of my shifters are trigger shifters so easy to open up and take out any broken remnants.Bear in mind that if the inner cable does break in the shifter the job possibly becomes an order of magnitude more complex and fiddly (and therefore expensive if you're subbing the work out to the LBS).