The short answer is no, applications for credit will not harm your TransUnion credit score, however, it may impact your eligibility for products.
When making an application for credit, lenders will check your credit report. When lenders check your credit report, this leaves a ‘Hard Search’ on your credit report. This search will then be visible to any other lenders assessing your application for a credit product and will inform them that you have recently applied for another credit product.
Other lenders will be able to see this search and know that you have had a recent application. Lenders won’t know if the application was successful or not but if they see multiple hard searches in a short period of time it could have an adverse effect on your eligibility, as lenders may see multiple hard searches as being desperate for credit.
If your application is rejected, don’t immediately go to the next provider and make another application. You should wait at least a month between applications, however for the best chance to be accepted it is worth waiting around 6 months between applications.