GPU in X16 slot running on X4?

If they are the same on both PCIe slots then yes, you could carefully remove the good clip from the other slot and attach it to the one you need it on. Just be very careful not to damage the board itself 

That said, if the system is working well and not showing any instability or other issues I wouldn't worry too much about it.

A GTX 780/R9 290 for example may suffer a performance hit but an R7 260/GTX 750 may not. The PCIe slots are rated as they are because of bandwidth. PCIe 2.0 x16 is what your GPU should be using but you are only getting x4 which means it is operating at a quarter of it's possible bandwidth. That isn't the whole story though, the speed you see is determined not only by the slot bandwidth but also by the speed of the device connected to it. If your GPU is mid to low end then PCIe x4 shouldn't be holding you back performance wise but if it is an upper mid range or faster GPU you will suffer more of a bottleneck the higher up the chain you go. 

Ultimately though PCIe 2.0 x4 would only bottleneck a high end GPU by at most 10fps in games. That is worst case scenario and would only really happen in titles where you are getting particularly high fps to begin with. If you have a game running at 30fps you may see 31fps or 32fps if you were not bottlenecked where a title running at 100fps may go up to 110fps if you had the full x16 bandwidth available to your GPU. Basically the bottleneck scales to the amount of throughput going through the slot to the CPU.

It should be PCIe 2.0 x16. I would try removing the graphics card and cleaning its contacts with some cotton and isopropyl alcohol. If you can, use a can of compressed air to blow out the slot from dust then make sure you seat the GPU properly in the slot and try again.

If it does not change then let us know.