FGW▸ don't operate from Portsmouth to Havant - but it's difficult to rub out a small link like that from the map.
Oh yes they do

... timetable 25 footnote:
In addition to the services shown, First Great Western also operate the following:
A Monday to Friday service departing Portsmouth Harbour at 0701 to Brighton, calling at Portsmouth & Southsea 0704, Fratton 0709, Havant 0719, Chichester 0731, Barnham 0739, Worthing 0756, Shoreham-by-Sea 0806, Hove 0814 and Brighton 0820
A Saturday service departing Portsmouth Harbour at 0648 to Brighton, calling at Portsmouth & Southsea 0653, Fratton 0657, Havant 0708, Chichester 0719, Barnham 0727, Worthing 0745, Shoreham-by-Sea 0757, Hove 0807 and Brighton 0815
A Sunday service departing Brighton at 2146 calling at Hove 2150, Shoreham-by-Sea 2156, Worthing 2203, Barnham 2218, Chichester 2226, Havant 2247, Fratton 2257, Portsmouth & Southsea 2300 and Portsmouth Harbour 2304
Edit to add smiley. Original post looked a bit gruff!