Can a 5V 1Ch Relay be switched with 4V?
You can avoid the issue by driving the relay on the low side. Then you will have the full swing, as long as the I/O current is enough. If you’re still unsure of the current output of the Arduino, you can consider boosting the current using a FET or bipolar transistor to drive the relay. 

Also, make sure to add a reverse-bias ‘freewheel’ diode across the coil. This absorbs the negative voltage spike when you remove power from the coil.

More: Your Arduino Pro 5V has a linear regulator on it to make internal 5V. If you’re feeding 5V as the raw supply (I’m guessing you are), the regulator will insert some dropout voltage, so internally the ATMega328 is seeing a lower voltage. You can overcome this in one of two ways: raise the raw supply to overcome the dropout, or bypass the regulator.

On the other hand, if, as suggested above, you drive the coil on the low side using a buffer (FET or bipolar transistor) then this becomes a non-issue.