From your post, i see you are using this MP3 kit with stock cassete player. At which point you are feeding the MP3 kit out in the stock player?
As from many posts, for powering these MP3 kit, mobile charger or LM7805 are available option among more.
Auto motive power rail is full of transient even it has battery (11~14V) voltage available.
Mobile charger has more ripples on putput and may produce noise in sound as some might have observed while powering 5V when powring MP3 kit from mobile charger.
LM7808 give ripple free +5V when powered from battery if use properly as long input is in specified range.
Vehicel power rails are full of transients and powering from this rail to DIY or other equipments that are not specially designed for automotive applications need carefull attention (In this case MP3 kit, Cheap!!!!!)
Please see the pis for voltage transient in vehcile power rail.
The vehicel power rail can go up more than 100V overshoot and clearlly this is not int he range of LM7805 (MAX 35V input).
For the equipemts designed for automotive applications , All have input filters that suppress these voltage transients.
If some one have opend audio player, then he can see big coil, capacitor and diode just after the interface conncetor. Definitly this is power filter for rest of electroncis.
So how to power to these like MP3 kits?
In the cassete player, you can take power from cassete player MOTOR (+9V ) point Then use LM7805 to get +5V.
This +9 V comes after poer filter.
Further more, this is switched power, So power only available when stock audio player is ON and cassete mode is selected.
You can put any dummy cassete there to turn ON cassete mode when cassete mode is selected from the button.
Who use cassetes these days? (not many!)
There is no reason for keeping on MP3 kit ON, when stock player is OFF (Off cource this is working as audio power amplifier and if it is OFF, then no need to keep MP3 kit ON.).
Keeping OFF MP3 kit while stock player is OFF also provide resume capability. (Power from the other place will keep running MP3 all the time when u start engine unless you turned it OFF manullay from the remote ) every time.
Hopes it helps.
Regards.