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The relay, there is something about that part that has me in two directions about it.
Please return the wire back to the original port on the Board for the relay the way it was in the first image you posted, and with the transistor still removed apply power.
As for the BJT NPN transistor, my thoughts on that are (if it turns out to be a problem) that even tho it has remained at the relays coil as sinking configuration, the flyback diode even tho adequate IN4148? or equivalent may not be properly clamped into the Solderless bread boards contacts, I have lost DMOS Fets that way (2N7000), the field and shear voltage levels are tremendous from even an 5v relay, the connection is not solid enough in the board, it can arc at the boards contact points resulting in quite a EMI field in the proximity of the connection point.
The FET damage here was of the Drain to Source being shorted in most cases resulting in an on at all times device, the Gate no longer had any control, an BJT would behave similar with excessive current being allowed to pass from the Base to the Emitter thus causing the 555IC to source more current than needed and generate heat.
Ok then, with the Transistor installed and the relay connected with the motor connected and no Base resistor connected.
Lets test the Transistor a bit more, using the 620R resistor, place the resistor from the Transistors base to the power positive rail, then apply power, the results should be the relay switches on, and the motor runs in that configuration as soon as powered on.
It's the boards design in general, Solderless bread boards have these problems regardless. even mine have these issues with inductive systems, Solderless bread boards are mainly just for small level low current logic switching devices such as Gates and LED drivers, An Comparator or an Op Amp can behave very oddly on these boards because of the Common loss issue, their requirement of stable voltage at the inputs make them unstable.
The 555 shows no direct issue with this as it is a charge discharge switch that utilizes an capacitor resistor charge as the time setting.
However the flyback issue can cause problems on these boards in many cases.
The NPN has shown to be damaged if the Relay does not "click" and the motor of coarse run when wired on that way.
Ok, By understanding the power requirement AND the effort, the issue can be with the relay at 9V, how?
because the relay at 5V is 71.4mA average, and at 9V it jumps to 128.5 mA through the coil, in other words the Transistors Base current requirement goes from 8mA to needing 13 or so mA current.
What has most likely happened is the relay at 9 volt exceeded current for the transistors Base requirement somewhere,
and the Transistor went out.
The relay needs an current limiting resistor at 9V! an 50R resistor at 1/2 Watt, would bring the coil current back down to 74mA, close to the coils preferred current.
The Transistors Base was not getting adequate current at some point to drive the exceeding current from the relay and was not switched on enough for the then 128.5 mA, and the also possibility of the inductive flyback was not being sent to the power rail positive enough to remove the voltage spike completely. the relay flyback diode placed over the relay to the right hand side rail positive to help improve the removal of the relay switch off spike.
In most boards I have parted out, any diodes in the flyback arrangement were pointed towards the power Source, that should not have been needed here.
If the Transistor is replaced, then the relay would require the 50R in series with the coil from power supply, the Transistors Base can still use the 620R from the 555 pin 3 at 9 v supply.
when the Collector pin is in the relays coil row, and the 620R resistor placed the same, to the power rail positive, apply power,
Does the relay click and the motor run?
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