Remove the speaker... get the system started first.
The LED RX on the Arduino Nano should light up if it gets data from the mp3 mini right? And if the mp3 mini worked, its led should also light up, or am I mistaken?
sth = something. It's the abbrevation I learned in school ,sorry if it's wrong, English is not my first language
I removed the speaker but it's still the same result
Unable to begin:
1.Please recheck the connection!
2.Please insert the SD card!
"light up" - yes, but at computer speeds... so maybe a brief flicker.
Okay. Remove all the wires from the SD card reader. You will get the same error. Verify that is the only error.
When you buy a Nano with header pins attached, they normally point downwards from the Nano, not upwards like yours appear to.
Did you put the header pins into the Nano like that? If so, did you solder them or just push them in?
I see two different Nanos now. Which one is in use and which is a stock photo?
They look the same to me. Are you confused with photo posted by @horace ?
the pins were loose in the package, I attached them by just pushing/ sticking them into the holes of the nano. Was that wrong?
Yes. Get 'em soldered!
And I would advise soldering them pointing down, you may want to use a breadboard or solder to a stripboard later.
it never flickers at the mp3 mini .
It's just the arduino nano now ,connected to my laptop. Yes, it's the same error. mp3 mini was not connected to the nano
I neither have the skills to solder nor the device for it 
And I doubt a breadboard or something like this will arrive till christmas evening. I thought the mp3 mini may just not be working as it's just damaged or so.
It's called a soldering iron. 
You need to solder the pins to the Nano before you can use it in a breadboard.
You should have bought a Nano with the pins already soldered if you don't have an iron.
you will have poor connections if any at all - the devices may operate intermittently or not at all
if you are not able to solder when you purchase modules such as the nano, DFPlayer, etc get them with pins already soldered in place
I didn't know that. Thanks
Do you know anyone nearby that has electronic soldering skills and equipment?
You could buy at least one Nano for that amount of money, with the pins already soldered.
There's a possibility that the MP3 DFmini is damaged now due to intermittent connections and powering itself via sneak paths on the signal lines.