The ingredients I used in my recipe were too cold and they did not mix well. Is there anything I can do to fix this

There are three points I'd like to suggest.

First, letting the mixture sit and come up to room temperature might help - especially if your recipe called for warmer ingredients. Or you might even look to set it someplace slightly warmer (near any mild heat source like you might need for dough) that might warm and soften your mixture a bit before trying to mix it again.

Second, as Dorothy mentioned, the lumpy texture may subside when mixed with the dry ingredients. One of the things that can work when smoothing lumps is to dry out the mixture enough to make a paste, and slowly loosen to a smooth liquid. It is possible that the dry ingredients will be enough to incorporate and smooth out any lumps in your mixture.

Third, and my favorite possibility - to misquote from a question about eggs - with an immersion blender and a sufficient amount of violence, there is much that is possible. If you have an immersion blender (or, yeah, even if you have a regular one - I just like the immersion ones better) - you can just blend the ingredients together, and count on the blender's mechanical thrashing, and possibly a bit of friction heat, to smooth your wet ingredient mix back down to smooth.

If the liquid in your bowl isn't very deep, an immersion blender might spray the batter around, true - but it actually doesn't take that much. If the blades of the blender are submerged (say, ~1 inch), an immersion blender can run for short bursts, if the whole head is under the batter (~1.5 inches) it can likely be blended for longer times without trying to fling batter around - and the bowl can be tilted for blending at the angle for the right thickness. I find it simpler to use and rinse the immersion blender in the bowl than fussing about transferring the batter to other containers to blend or mix in.

Although while I prefer immersion blenders, you might also try a regular blender, or a mixer, or even a milk frother (depending on amount, power, and viscosity of your mixture) to beat the lumps out of the mix and leave it nice and frothy. You might, if the liquid is thin and the lumps are stiff, slowly start adding some of the dry ingredients to thicken the mixture (the lumps blend better the closer everything is to the same texture), and keep blending everything smooth.

In any case, once the mix has been beaten down into smoothness, you can continue your recipe, folding in the rest of the dry ingredients and continuing from there.

Good luck, I hope it all turns out well! :)

In this situation, just continue making it.

Butter emulsions are very sensitive, and it can happen that they don't build properly. But the lumps you have there are only butter, not flour. A very discerning eater might notice the difference of lumpy-butter cake after the cake is baked, but we are talking the difference between "great" and "sublime" here. Chances are, the cake will be good enough for you.

If you have a high speed blender, you might try it, as Megha suggested. If you have a standard one, it probably won't work well, so not worth the trouble.