The pattern you described is fairly common. You as the US (or Canadian) national developed a romantic long distance relationship (LDR) with a Brit, and sought to maintain the relationship by using your status as a non-visa national to visit without a visa. Last year you were served removal papers and got a temporary admission of 1 day. We don't have the IO's transcript of your landing interview, but typically those situations involve some cat-and-mouse type of interaction that gets them more upset than normal.
Then you applied for entry clearance (a good strategy), but were refused.
They got you on V 4.2 (a+b+c), which is symptomatic of a person whose history indicates they are not a genuine visitor and should be using one of the other inward routes (a fiance visa for example). LDR's are fine, but the person needs to be leading a properly integrated lifestyle in their home country. They expect that during one of your visits you will likely go underground and engage in further abuse of the rules.
Some of your questions...
Will I be refused again?
We're not prophets, but it can be difficult to maintain an LDR after they have introduced V 4.2 (b) as a grounds. It means you have worn out your welcome and your visits suggest a further agenda (one characterised by abuse).
Is there a way to check notes or restrictions associated with my
Yes, in general the UK has a procedure called Subject Access Request where they will release your records in a very limited form. It will not, for example, contain the transcript behind your earlier removal, and it will not include any information that might help you evade immigration controls.
Is there a way to clear any such restrictions associated with me?
Yes, the absolute golden solution is to get an entry clearance. This has the effect of wiping the slate to the extent that you have successfully established yourself as a genuine visitor who is using the rules lawfully. I recognise that just having been refused, this solution seems like a cruel tautological recursion. But as the Brits say, that's just "hard cheese".
Can I write a formal letter to the UK Home Office to have a chance to
Of course. You can even petition HM (she is contactable at Buckingham Palace, via an assistant) or write to the Home Secretary. In either case you will get back a standard letter informing you that you are free to make a fresh application for entry clearance. In other words, do not expect very much. Entry Clearance Officers have a mandate from the government to make decisions on behalf of the Home Secretary and that's that. Visitor applications do not have appeal rights.
The most frustrating part about all of this, is I am not, nor have I
ever had any intention of trying to cheat the system or do anything
There is a lengthy history of abuse with people in LDR's. They lie to the IO's, they waste time playing cat-and-mouse, they overstay, they tie up the courts, they evade fees and other requirements. So much so that they have been trying to contain it with rule changes for a long time (more than 20 years). So while your personal intentions are an unknown factor, you clearly fit the pattern and that's what they work from.
I am beyond heartbroken because I think if I were to try and fly back
to see my significant other again, I would be refused again. I don't
know what to do. We both don't know what to do. All I want to do is
spend time with the person I love. I'm not trying to do anything
illegal. I just want to be with them.
"Significant other", "partner", "the person I love", this is language that describes a serious commitment. As I mentioned above one of the settlement routes is more appropriate. They may be adjusting the financial hurdles following a court decision last week. In addition to fiance or spouse, you can also try the student route.
Any help, options or advice would be greatly appreciated. More than
There are several immigration solicitors of national stature who specialise in entry clearance for LDR's. I took a law course in it a while back (and know the instructor personally for about 23 years) and it was very well attended. Use the search pages at ILPA.
Comparisons. The OP has asked her questions on two internet resources. It's interesting to see how the same case was handled...
- For comparative purposes, the interested reader is invited to view
the OP's thread at UKY. It's a forum dedicated to American women
trying to marry Brits and lots of their advice has sound
practicality. They are not legal professionals, their very
impressive experience derives from tens-of-thousands of threads, so
always double check their stuff to make sure.
- Also for comparison purposes, the OP published her thread on
Citation from the study that I linked to above...
...the indication is that many of these persons had intended to marry
all along but had not obtained leave to enter on this basis and had
therefore lied about their intentions to the entry clearance officer...
And from the seminal Home Office Online Report ...
Some IOs were surprised, and suspicious, when people travel to spend
time with someone they hardly know or have never met. If people are
going to visit someone in another country, IOs assume they will have
had recent or regular contact and will know something about the other
person. Internet relationships attract particular attention.