CD-R's audio jumps on a RadioShack 3905 CD walkman

What brand CD-Rs do you use? Certain brands like Memorex are hit and miss due to how cheap they are. In addition to that, some old CD players only like 650MB “Redbook” blanks, and do not work well with 700MB blanks (the problem is this 700MB media marginally violates the Redbook spec, but are Redbook/Orange Book complaint if you burn 650MB or less). The manual date code linked by @jayeff leads me to think it should be new enough to reasonably handle 700MB burned discs, but some older cheap CD players like yours are still mildly problematic. On equipment from the 80s-90s it tends to totally reject burned discs - even some older car radios like some of the older Toyota radios used in the VX10 chassis, or pre-W202 Mercedes radio have problems or want 650MB media only and struggle with 700MB discs. If it's incompatible with 700MB blanks, you can use 650MB blanks. However, the irony is they used to be cheap but now they're more expensive by a good margin because what's left is what is left since they are no longer manufactured. I'd recommend a better player which isn't sensitive to the spec violation.

Failing a 700MB CD-R incompatibility, try using a quality CD-R (non pro) like Verbatim, Maxell or Imation. With the exception of Staples, avoid store brands. If it’s still an issue, look for 700MB “pro” blanks like Ritek Pro and Japan Disc Taio Yudens (GRADE A - no VAL which are Grade B), or Grade A TY CMC PRO media. These are built the same but have a much lower BLER rate and tend to behave more often in problem child machines. In addition, if I find the pro discs are as much as consumer media I usually buy it given a choice every time even for cheap players just because of the far improved reliability.

TIP: In some cases, a CD ripper like EAC that burns discs as well is better then general purpose writing software. EAC can better replicate commercial TOC mapping, which may help. Try using EAC before writing off the player as burned disc incompatible.

If it isn’t a blank issue, some CD players just do not handle recordable discs well no matter what. You can even use gold archive discs all day long, and continue to have issues with these machines.

The only caution I have here is to avoid no-name Chinese media, like Tiger or non-branded media. The discs are prone to high failure rates, and have horrible BLER rates. China does not care - they change or remove the branding! Memorex is also best avoided for production use (their cassettes were usually slightly better then Type 0 garbage) and failure prone. It doesn’t suffer from the BLER rate as much as Chinese media but it’s cheap enough it's not as good as a quality consumer blank like Verbatim. Quality equipment can cope better.

The issue with cheap CD-Rs in some of this legacy equipment is a lot of discs are Grade A now (this was not always the case, as early CD-Rs were usually A/B/C mixed), but the reason why they’re sold as consumer media - the BLER rate is high enough it isn’t acceptable for “professional” use, but it’s acceptable for consumer use.