This is a statement which relates to epistemology. This basically means a theory of knowledge with regards to its validity. 
Let us break down the statement. “Irrationally held truths” means something that is probably factual but lacks proof that this truth is based on evidence and hence, cannot be justified. By “reasoned errors”, Huxley is trying to say that if an individual has wrongly reasoned something, their logic is open to review for other researchers and scientists who can correctly give the accurate reasoning. 

Huxley, in this statement, is trying to convey the message that ‘something that is probably factual but lacks evidence’ is more dangerous than ‘something that is wrongly reasoned but is open to criticism’. 

With regards to medicine, there are and have been many cases where the treatment given to a patient lacked evidence to prove that it is effective. Homeopathy is classic example of this. It is widely regarded as ‘natural medicine’, mainly in the eastern part of the world. They are not yet proven to be effective by science. However, some people do claim that it has worked for them and hence, should work for everyone. Such claims could prove to be fatal for some patients. Homeopathy can probably just have a placebo effect on some people but it is not necessary that it will do the same on others. Therefore, some may argue that it is inappropriate to treat someone with a serious illness with homeopathy since there is no evidence that it is effective. 

The contrary argument could be that ‘it is better to arrive at the correct answer without any method or incorrect method than to arrive at a wrong answer using the correct method’. This would be preferable in many cases in medicine. Some would argue that it is better to treat a patient with an invalid method with little evidence if no valid treatments work for them. 
An example of this is the Deep Brain Stimulation to treat Parkinson’s disease. Even though we do not know the exact mechanism behind this treatment, we can see positive results in patients. Therefore, the argument here is that as long as treatments repeatedly produce positive results on patients, we do not really need to know how they work. 

In order to turn something irrational to rational, scientific researches need to be carried out. This allows evidence to support any claim. The work of many scientists in the past had been disregarded because there were other ‘explanations’ for the phenomena occurring around us e.g. religious explanations. However, later on, those works of scientists were later found to be true using rigorous and logical scientific experiments and studies.