We may not know much about how the brain works at the microscopic or macroscopic level. But we do know a few things.
The brain is composed of brain cells connected to many other brain cells. One brain cell fires, stimulating others. If this happens between several cells, a signal of some sort has been passed. At the same time the cells adapt subtly, so the next time the first cell fires it is more likely to do the same thing.
It is this sort of network that machine learning models aim to copy. For experimental reasons we generally separate the response (the passing of the signal) from the training (the adaption of the network). This lets us run experiments on our neural net models which are almost impossible to do with real neural nets.
Our vision system has two main pathways - the ventral stream is associated with identifying and recognising objects, and has access to our long-term memory, and the dorsal stream is more concerned with spatial relationships between objects, and with muscular actions and coordination, and has no obvious access to the long-term memory. But both streams have 'neuroplasticity' - they will react to the signal they pass, and can adapt to change and recover from damage. Memory is everywhere. This makes it hard to remove a memory completely.
In the very short term, we are continuously forgetting things. If a clock strikes, and someone asks what the time is, we can could remember what we heard, count the strikes, and say what time it was. A minute or so later, we may have forgotten that we had ever heard the clock strike.
If something has our attention, then the memory gets harder to erase. If you think "what did I just see/hear?" then there will be some conscious memory. This is not yet the long-term memory. If we do not use the information, it will probably be lost. This seems to be a bit of what anaesthetics do. But if it was a significant memory and you are not on drugs then you will remember it, even if you would rather not. Experience tells us that we can modify our memories. Police interrogation can change witness memories. Peer pressure can cause us to doubt ourselves. But this is more like rubbing out parts of a drawing drawn with an H pencil rather than a 3B, and putting something else over the top.
If we were a neural net model, we might be able to locate the particular set of weights at a node that (say) recognised our ex, and disable them. In a brain, this is much harder, because the experiments necessary to find the cell (or probably many cells) that do this would reinforce the memories you want to erase. This does not mean it is impossible, for therapy is supposed to do this sort of thing. It may be more efficient, if assisted by dream image reconstruction from MRI (a real thing), and drugs, and a better understanding of how to go about this sort of thing. But it is unlikely to be a 'Men in Black' flash erasure if the memory is older than a minute or so.